Postulation of general quartuple fat point schemes in

arXiv:1103.5317v1 [math.AG] 28 Mar 2011 POSTULATION OF GENERAL QUINTUPLE FAT POINT SCHEMES IN P3 E. BALLICO, M. C. BRAMBILLA, F. CARUSO, AND M. SALA Abstract. We study the postulation of a general union Y of double, triple, quartuple and quintuple points of P3 . In characteristic 0, we prove that Y has good postulation in degree d ≥ 11. The proof is based on the combination of the Horace differential lemma with a computer-assisted proof. We also classify the exceptions in degree 9 and 10. 1. Introduction Let K be a field of characteristic 0, n ∈ N and Pn = Pn (K). In this paper we study the postulation of general fat point schemes of P3 with multiplicity up to 5. A fat point mP is a zero dimensional subscheme of P3 supported on a point P and with (IP,P3 )m as its ideal sheaf. A general fat point scheme Y = m1 P1 +. . .+mk Pk , with m1 ≥ . . . ≥ mk ≥ 1 is a general zero-dimensional scheme such that its support Yred is a union of k points and for each i the connected component of Y supported on Pi is the fat point mi Pi . We say that the multiplicity of Y is the maximal multiplicity, m1 , of its components. Studying the postulation of Y means to compute the dimension of the space of hypersurfaces of any degree containing the scheme Y . In other words this problem is equivalent to computing the dimension δ of the space of homogeneous polynomials of any degree vanishing at each point Pi and with all their derivatives, up to multiplicity mi − 1, vanishing at Pi . We say that Y has good postulation if δ is the expected dimension, that is, either the difference between the dimension of the polynomial space and the number of imposed conditions or just the dimension of the polynomial space (when δ would exceed it). This problem was investigated by many authors in the case of P2 , where we have the important Harbourne-Hirschowitz conjecture (see [7] for a survey). In the case of Pn , for n ≥ 2, the celebrated Alexander-Hirschowitz theorem gives a complete answer in the case of double points, that is when mi = 2 for any i ([1, 2], for a survey see [5]). For arbitrary multiplicities and arbitrary projective varieties there is a beautiful asymptotic theorem by Alexander and Hirschowitz [3]. Here we focus on the case of general fat point schemes Y ⊂ P3 . In this case a general conjecture which characterizes all the general fat point schemes not having good postulation was proposed by Laface and Ugaglia in [12]. The good postulation of general fat point schemes of multiplicity 4 was proved for degrees d ≥ 41 in [4] by the first two authors. Then Dumnicki made a real breakthrough. In particular he showed, in [10], how to check the cases with degree 9 ≤ d ≤ 40. Stimulated by his 1991 Mathematics Subject Classification. 14N05; 15A72; 65D05. Key words and phrases. polynomial interpolation; fat point; zero-dimensional scheme; projective space. 1 2 E. BALLICO, M. C. BRAMBILLA, F. CARUSO, AND M. SALA results, we consider now the case of fat point schemes of multiplicity 5 and we solve completely the problem of the good postulation. Indeed we prove the following theorem. Theorem 1. Let P3 = P3 (K), where K is a field of characteristic 0. Fix nonnegative integers d, w, x, y, z such that d ≥ 11. Let Y ⊂ P3 be a general union of w 5-points, x 4-points, y 3-points and z 2-points. Then Y has good postulation with respect to degree-d forms. The more natural way to prove our result would be to adopt a usual two-parts proof: we might prove the theorem for d ≥ 66 with the same theoretical approach as in [4] and then we might prove the remaining finite cases with the computer. We do not follow this consolidated path, because the computer calculations at level d ≥ 60 are infeasible with nowadays means. Instead, the proof of our result is an innovative combination of computer computation and theoretical argument, as in the following logical outline: a) First, we prove Theorem 1 for degrees d = 11 using our servers (Th. 16). b) Second, we improve the argument of [4] and so we are able to prove Theorem 1 for degrees d ≥ 53, with a theoretical proof depending on both known results (Remark 13) in the case of fat points of P2 and on a) (d = 11). This is presented in Section 3. c) Then, we perform several computer calculations (Lemma 15). d) Then, we give a theoretical proof that restricts the required computations for the remaining cases (11 ≤ d ≤ 52) to some feasible jobs. This proof depends on the previous computational results. The main point here is that an iterated use of some results by Dumnicki ([9, 10]) allows us to greatly reduce the number of cases to be considered, by adding points of higher multiplicity. In particular we make use of points of multiplicity 10 and 13. Another tool we use is a result concerning low degrees and few quintuple points (see Proposition 23). This result is proved by a modification of the general proof contained in Section 3 and indeed allows us to exclude many cases from the explicit checking by computer. All this is reported in Section 4. e) Finally, we perform direct computer checks for the surviving cases, as detailed in Section 5. Our computer calculations are deterministic and produce several digital certificates, that allow any other researcher to verify our results precisely. They rely on the efficient software package MAGMA ([15]), whose linear algebra over finite fields outperforms any other software that we tried. All our programmes and their digital certificates are publicly accessible at http://www.science.unitn.it/~sala/fat_points/ In the remainder of the paper we provide two sections, as follows. In Section 6 we classify all the exceptions arising in degree 9 and 10 (relying again on a computer-aided proof). It turns out that, in these cases, the Laface-Ugaglia conjecture is true. In Section 7 we collect several remarks on our results and their consequences. QUINTUPLE POINTS 3 2. Preliminaries In this section we fix our notation (which is the same as in [4] whenever possible), prove several preliminary results and summarize our computational results. Let Pn be the projective space on a field K, with char(K) = 0 and n ∈ N. Note that we do not assume that K is algebraically closed. However, some of the references which we will use assume that the base field is algebraically closed. In the next lemma we explain why we are allowed to use these results. Lemma 2. Let K denote the algebraic closure of K. Fix non-negative integers n, d, x, y, z, w, s such that n ≥ 1. Assume that a general disjoint union of w quintuple points, x quartuple points, y triple points, z double points and s (simple) points in Pn (K) has good postulation in degree d, i.e. either



n+3 n+2 + (n + 1)z + s ≥ n+d • h0 (Pn (K), IZ (d)) = 0 and n+4 4 w
+ 3 x
+ 2 y n

n+3 n+2 n+d • or h1 (Pn (K), IZ (d)) = 0 and n+4 4 w+ 3 x+ 2 y+(n+1)z+s ≤ n . Then there is a disjoint union W of w quintuple points, x quartuple points, y triple points, z double points and s points in Pn (K) with good postulation in degree d.


n+3 n+2 Proof. Increasing s, if necessary, we reduce to the case n+4 4 w+ 3 x+ 2 y+
(n + 1)z + s ≥ n+d n . Let ν = w + x + y + z + s. Let E be the subset of n ν P (K) parametrizing all the ν-ples of distinct points of Pn (K). For any A ∈ E, let ZA ⊂ Pn (K) be the fat point subcheme of Pn (K) in which the first w (resp. x, resp. y, resp. z, resp. s) fat points share multiplicity 5 (resp. 4, resp. 3, resp. 2, resp. 1) and (ZA )red is the set associated to A. By semicontinuity there is a non-empty open subset U of E such that for all A ∈ U (K) we have h0 (Pn (K), IZA (d)) = 0. nν Since K is infinite, Knν is dense in K . Hence Pn (K)ν is Zariski dense in Pn (K)ν . Thus, there is B ∈ U (K) such that the scheme ZB satisfies h0 (Pn (K), IZB (d)) = 0 and it is defined over K.  From now on, K is any field with char(K) = 0 and Pn = Pn (K). For any smooth n-dimensional connected variety A, any P ∈ A and any integer m > 0, an m-fat point of A (or just m-point) {mP, A} is defined to be the (m−1)-th infinitesimal neighborhood of P in A, i.e. the closed subscheme of A with (IP,A )m as its ideal sheaf. As a consequence, {mP, A}red = {P } and the length of {mP, A}
is length({mP, A}) = n+m−1 . To ease our notation, we will write mP instead n of {mP, A} when the space A is clear from the context, and mostly we will have A = Pn for n = 2, 3. We call general fat point scheme of A (or general union for short) any union Y = m1 P1 + . . . + mk Pk , with P m1 ≥ . . . ≥ mk ≥ 1, and P1 , . . . , Pk general points of Pn . We denote deg(Y ) = length(mi Pi ). Given a positive integer d, we will say that a zero-dimensional scheme Y of Pn has good postulation in degree d if the following conditions hold:
1 n (a) if deg(Y ) ≤ n+d n
, then h (P , IY (d)) = 0, n+d 0 n (b) if deg(Y ) ≥ n , then h (P , IY (d)) = 0. We will also use the notation Ln (d; m1 , . . . , mk ) for the linear system of hypersurfaces of degree d in Pn passing through a general union Y = m1 P1 + . . . + mk Pk . The virtual dimension of L = Ln (d; m1 , . . . , mk ) is   n+d vdim(L) = − deg(Y ) − 1 n 4 E. BALLICO, M. C. BRAMBILLA, F. CARUSO, AND M. SALA and the dimension of the linear system always satisfies dim(L) ≥ vdim(L). We say that L is special if dim(L) > max{vdim(L), −1}. It is easy to see that a linear system L is special if and only if the corresponding general union does not have good postulation in degree d. For more details we refer to [7]. Remark 3. Let do ≥
2. Assume that Y is any general fat point scheme in Pn such 0 that deg(Y ) ≥ n+d . If we know that Y has good postulation in degree d ≥ d0 , n we can claim that Y has good postulation in any degree, as follows. For d ≥ d0 , there is nothing to prove. Since, for any d ≥ 1, there is an injective map H 0 (Pn , IY (d − 1)) ֒→ H 0 (Pn , IY (d)) , then h0 (Pn , IY (d)) = 0 implies h0 (Pn , IY (d − 1)) = 0. But h0 (Pn , IY (d0 )) = 0 and so h0 (Pn , IY (d)) = 0 for any d < d0 , which proves that Y has good postulation. Similarly, if h0 (H, IY ∩H (d0 )) = 0, then h0 (H, IY ∩H (d0 − 1)) = 0. The following general lemma will be useful in the sequel. Lemma 4. Let Σ be an integral projective variety on K and let L be a linear system (not necessarly complete) of divisors on Σ. Fix an integer m ≥ 1 and a general point P ∈ Σ. Let L(−mP ) be the sublinear system of L formed by all divisors with a point of multiplicity at least m at P . Then we have dim(L(−mP )) ≤ max{dim(L) − m, −1}, and, for any 1 ≤ k ≤ m, dim(L(−mP )) ≤ max{dim(L(−kP )) − (m − k), −1}. Proof. The case m = 1 is obvious. We assume by induction that dim(L(−(m − 1)P )) ≤ max{dim(L) − m + 1, −1}. By [6, Proposition 2.3] it follows that dim(L(−mP )) ≤ max{dim(L(−(m − 1)P )) − 1, −1} , and so we get the desired inequality. The proof of the second inequality is analogous.  In the following lemma we show that in order to prove Theorem 1 for all quadruples (w, x, y, z) of non-negative integers it is sufficient to prove it only for a small set of quadruples (w, x, y, z). Lemma 5. Fix an integer d > 0. For any quadruple of non-negative integers (w, x, y, z), let Y (w, x, y, z) ⊂ P3 denote a general union of w 5-points, x 4-points, y 3-points and z 2-points. If Y (w, x, y, z) has good postulation in degree d for any quadruple (w, x, y, z) such that     d+3 d+3 − 3 ≤ 35w + 20x + 10y + 4z ≤ +∆ 3 3 where  13 if w > 0 and x = y = z = 0,    8 if x > 0 and y = z = 0, (1) ∆= 4 if y > 0 and z = 0,    1 if z > 0 then any general quintuple fat point scheme has good postulation in degree d. QUINTUPLE POINTS 5
Proof. If a quadruple (w, x, y, z) is such that 35w + 20x + 10y + 4z ≤ d+3 − 4, then 3 ′ we want to prove that
h1 (Pn , IY (d)) = 0, where Y = Y (w, x, y, z). Let z > 0 be the
d+3 d+3 ′ integer such that 3 − 3 ≤ 35w + 20x + 10y + 4z + 4z ≤ 3 . By hypothesis we know that Y ′ = Y (w, x, y, z + z ′ ) has good postulation, that is, h1 (P3 , IY ′ (d)) = 0. Since Y ⊂ Y ′ , then it is easy to see that h1 (P3 , IY (d)) ≤ h1 (P3 , IY ′ (d)) = 0, and so Y has good postulation.
Now assume that w > 0, x = y = z = 0, and 35w ≥ d+3 + 14. Let Y be the 3 corresponding general union of w quintuple points. This time we want to
prove
that h0 (P3 , IY (d)) = 0. Let w′ > 0 such that d+3 − 21 ≤ 35(w − w′ ) ≤ d+3 + 13. 3 3 Now we consider the following subcases:
′ ′ ′ • if 35(w−w′ ) ≥ d+3 3 , then we take the union Y = Y (w−w , 0, 0, 0) of w−w ′ quintuple general points. Since we can assume Y ⊂ Y , we immediately have h0
(P3 , IY (d)) ≤ h0 (P3 , IY ′ (d))
= 0, and so Y has good postulation. • If d+3 − 5 ≤ 35(w − w′ ) ≤ d+3 − 1, we take the union Y ′ = Y (w − 3 3 ′ ′ w , 0, 0, 0) of w − w quintuple general points. Since Y contains at least one further quintuple point, we can consider Y ′′ = Y (w − w′ + 1, 0, 0, 0) and we can assume that Y ′ ⊂ Y ′′ ⊆ Y . Note that Y ′ has good postulation by hypotesis, and h0 (P3 , IY ′ (d)) ≤ 5. Hence by Lemma 4 we have h0 (P3 , IY ′′ (d)) ≤ max{h0 (P3 , IY ′ (d)) − 5, 0} = 0. Then we have that Y ′′ has good consequently Y has good postulation.
postulation, and
d+3 ′ • If d+3 − 12 ≤ 35(w − w ) ≤ − 6, then we take Y ′ = Y (w − w′ , 0, 1, 0), 3 3 ′ i.e. a
general union of w − w
quintuple points and one triple point. Now d+3 − 2 ≤ deg(Y ′ ) ≤ d+3 + 4 and by hypothesis Y ′ has good postula3 3 ′ tion. Since we can assume Y ⊂ Y , by Lemma 4 we have h0 (P3 , IY (d)) ≤ max{h0
(P3 , IY ′ (d)) − 2, 0} = 0, and so
Y has good postulation. • If d+3 − 21 ≤ 35(w − w′ ) ≤ d+3 − 11, then we take Y ′ = Y (w − 3 3

d+3 ′ ′ w , 1, 0, 0). Now 3 − 2 ≤ deg(Y ) ≤ d+3 + 9 and by hypothesis Y ′ 3 has good postulation. Since we can assume Y ′ ⊂ Y , by Lemma 4 we have h0 (P3 , IY (d)) ≤ max{h0 (P3 , IY ′ (d)) − 1, 0} = 0, and so Y has good postulation.
Assume now x > 0, y = z = 0 and 35w + 20x ≥ d+3 + 9. Let Y = Y (w, x, 0, 0) 3 be the corresponding general union and we want to prove that h0 (P3 , IY (d)) = 0.
d+3 If 35w ≥ 3 , then Y ′ = Y (w, 0, 0, 0) has good postulation by the previous step and clearly it follows
that Y has good postulation.
Otherwise there exists 0 < x′ < x such that d+3 − 11 ≤ 35w − 20(x − x′ ) ≤ d+3 + 8. 3 3 Now we consider the following subcases:
′ ′ • If 35w−20(x−x′) ≥ d+3 3 −4, then we take the union Y = Y (w, x−x , 0, 0). Since Y contains at least one further quartuple point, by Lemma 4 we have h0 (P3 , IY (d)) ≤ max{h0 (P3 , IY ′ (d)) − 4, 0} = 0, and so Y has good postulation.

• If d+3 − 6 ≤ 35w + 20(x − x′ ) = d+3 − 5, then we take Y ′ = Y (w, x − 3 3

d+3 ′ ′ x , 0, 1) and we have 3 − 2 ≤ deg(Y ) ≤ d+3 − 1 and by hypothesis 3 Y ′ has good postulation. Since we can assume Y ′ ⊂ Y , by Lemma 4 we have h0 (P3 , IY (d)) ≤ max{h0 (P3 , IY ′ (d)) − 2, 0} = 0, and so Y has good postulation. 6 E. BALLICO, M. C. BRAMBILLA, F. CARUSO, AND M. SALA

• If d+3 − 11 ≤ 35w + 20(x − x′ ) ≤ d+3 − 7, then we take Y ′ = Y (w, x − 3 3

d+3 x′ , 1, 0) and we have 3 − 1 ≤ deg(Y ′ ) ≤ d+3 + 3 and by hypothesis 3 Y ′ has good postulation. Since we can assume Y ′ ⊂ Y , by Lemma 4 we have h0 (P3 , IY (d)) ≤ max{h0 (P3 , IY ′ (d)) − 1, 0} = 0, and so Y has good postulation.
Now assume y > 0, z = 0 and 35w + 20x+ 10y ≥ d+3 + 6. Let Y = Y (w, x, y, 0) 3 be the corresponding general union and we want to prove that h0 (P3 , IY (d)) = 0.
d+3 ′ If 35w + 20x ≥ 3 , then Y = Y (w, x, 0, 0) has good postulation by the previous steps and clearly it follows that Y has good postulation. Otherwise there

d+3 ′ exists 0 < y ′ < y such that d+3 − 4 ≤ 35w − 20x + 10(y − y ) ≤ + 5. 3 3 Now we consider the following subcases:
• If 35w − 20x + 10(y − y ′ ) ≥ d+3 − 3, then we take the union Y ′ = 3 ′ Y (w, x, y − y , 0). Since Y contains at least one further triple point, by Lemma 4 we have h0 (P3 , IY (d)) ≤ max{h0 (P3 , IY ′ (d)) − 3, 0} = 0, and so Y has good postulation.
′ ′ • If 35w−20x+10(y −y ′) = d+3 3
−4, then we take Y = Y (w, x, y −y , 1) and d+3 ′ ′ we have that deg(Y ) = 3 and by hypothesis Y has good postulation. It immediately follows that h0 (P3 , IY (d)) ≤ h0 (P3 , IY ′ (d)) = 0, and so Y has good postulation.
Finally assume that z > 0 and 35w + 20x + 10y + 4z ≥ d+3 + 2. Let 3 Y = Y (w, x, y, z) be the corresponding general union and we want to prove that h0 (P3 , IY (d)) = 0.
′ If 35w + 20x + 10y ≥ d+3 3 , then Y = Y (w, x, y, 0) has good postulation by the previous steps and clearly it follows
that Y has good postulation. Otherwise
there d+3 ′ exists 0 < z ′ < z such that d+3 − 2 ≤ 35w − 20x + 10y + 4(z − z ) ≤ + 1. 3 3 Now we take the union Y ′ = Y (w, x, y, z − z ′ ). Since Y contains at least one further double point, by Lemma 4 we have h0 (P3 , IY (d)) ≤ max{h0 (P3 , IY ′ (d)) − 2, 0} = 0, and so Y has good postulation.  Remark 6. Lemma 4 and Lemma 5 heavily use char(K) = 0, but they will be useful also in Section 5. Given a general fat point scheme Y of Pn and a hyperplane H ⊂ Pn , we will call trace of Y the subscheme (Y ∩H) ⊂ H and residual of Y the scheme ResH (Y ) ⊂ Pn with ideal sheaf IY : OPn (−H). Notice that if Y is an m-point supported on H, then its trace Y ∩H is an m-point of H and its residual ResH (Y ) is an (m−1)-point of Pn . We will often use the following form of the so-called Horace lemma. Lemma 7. Let H ⊂ Pn be a hyperplane and X ⊂ Pn a closed subscheme. Then h0 (Pn , IX (d)) ≤ h0 (Pn , IResH (X) (d − 1)) + h0 (H, IX∩H (d)) h1 (Pn , IX (d)) ≤ h1 (Pn , IResH (X) (d − 1)) + h1 (H, IX∩H (d)) Proof. The statement is a straightforward consequence of the well-known Castelnuovo exact sequence 0 → IResH (X) (d − 1) → IX (d) → IX∩H (d) → 0. For more details see e.g. [5, Section 4].  QUINTUPLE POINTS 7 The basic tool we will need is the so-called Horace differential lemma. This technique allows us to take a differential trace and a differential residual, instead of the classical ones. For an explanation of the geometric intuition of the Horace differential lemma see [3, Section 2.1]. Here we give only an idea of how the lemma works. Let Y be an m-point of Pn supported on a hyperplaneH ⊂ Pn . Following the language of Alexander and Hirschowitz, we can describe Y as formed by infinitesimally piling up some subschemes of H, called layers. For example the layers of a 3-point {3P, Pn } are {3P, H},{2P, H}, and {P, H}. Then the differential trace can be any of these layers and the differential residual is a virtual zero-dimensional scheme formed by the remaining layers. In this paper we will apply several times the following result which is a particular case of the Horace differential lemma (see [3, Lemma 2.3]). Lemma 8 (Alexander-Hirschowitz). Fix an integer m ≥ 2 and assume that char(K) = 0 or char(K) > m. Let X be an m-point of Pn supported on P and H ⊂ Pn a hyperplane.Then for i = 0, 1 we have hi (Pn , IX (d)) ≤ hi (Pn , IR (d − 1)) + hi (H, IT (d)) where the differential residual R and the differential trace T are virtual schemes of the following type: m T R 2 {P, H} {2P, H} (1, 3) 3 {P, H} ({3P, H}, {2P, H}) (1, 6, 3) 3 {2P, H} ({3P, H}, {P, H}) (3, 6, 1) 4 {P, H} ({4P, H}, {3P, H}, {2P, H}) (1, 10, 6, 3) 4 {2P, H} ({4P, H}, {3P, H}, {P, H}) (3, 10, 6, 1) 4 {3P, H} ({4P, H}, {2P, H}, {P, H}) (6, 10, 3, 1) 5 {P, H} ({5P, H}, {4P, H}, {3P, H}, {2P, H}) (1, 15, 10, 6, 3) 5 {2P, H} ({5P, H}, {4P, H}, {3P, H}, {P, H}) (3, 15, 10, 6, 1) 5 {3P, H} ({5P, H}, {4P, H}, {2P, H}, {P, H}) (6, 15, 10, 3, 1) 5 {4P, H} ({5P, H}, {3P, H}, {2P, H}, {P, H}) (10, 15, 6, 3, 1) In the previous lemma we described the possible differential residuals by writing the subsequent layers from which they are formed. These layers are obtained by intersecting with the hyperplane H many times. In particular the notation e.g. R = ({3P, H}, {2P, H}) means that R∩H = {3P, H} and ResH (R)∩H = {2P, H}, and, finally, ResH (ResH (R)) ∩ H = ∅, the latter equality being equivalent to ResH (ResH (R)) = ∅, because Rred ⊂ H. Moreover, for each case in the statement we write in the last column the list of the lengths of the fat points of H that we will obtain intersecting many times with H. Throughout the paper, when we will apply Lemma 8, we will specify which case we are considering by recalling this sequence of the lengths. For example, if we apply the first case of Lemma 8, we will say that we apply the lemma with respect to the sequence (1, 3). The next two arithmeticals lemma will be used in the sequel. Lemma 9. Let w, x, y, z be non negative integers such that   d+3 35w + 20x + 10y + 4z ≤ + 13. 3 Let α = ⌊ 2x+y 42 ⌋ and assume that w ≤ α − 1. Then 35w ≤
. 1 d+3 12 3 8 E. BALLICO, M. C. BRAMBILLA, F. CARUSO, AND M. SALA
Proof. By hypothesis we have 20x + 10y ≤ d+3 + 13, from which we have 3     13 1 d+3 1 d+3 20x + 10y + . −1≤ −1≤ w ≤α−1≤ 3 3 420 420 420 420  Lemma 10. Fix non-negative integers t, a, b, c, u, v, e, f, g, h such that t ≥ 18,   t+2 (2) 15a + 10b + 6c + 3u + v + 10e + 6f + 3g + h ≤ 2 and (e, f, g, h) is one of the following quadruples: (0, 0, 0, 0),(0, 0, 0, 1), (0, 0, 0, 2), (0, 0, 1, 0), (0, 0, 1, 1), (0, 0, 1, 2), (0, 1, 0, 0),(0, 1, 0, 1), (0, 1, 0, 2), (0, 1, 1, 0), (1, 0, 0, 0), (1, 0, 0, 1), (1, 0, 0, 2), (1, 0, 1, 0), (1, 0, 1, 1). Then the following inequality holds:   t+1 10a + 6b + 3c + u + 15e + 15f + 15g + 15h ≤ (3) . 2 If e + f + g + h ≤ 2, then the statement holds for any t ≥ 15. If e = f = g = h = 0, then the statement holds for any t ≥ 4. Proof. By contradiction, let us assume that (4)   t+1 10a + 6b + 3c + u + 15e + 15f + 15g + 15h > , 2 which, together with (2), implies (5) 5a + 4b + 3c + 2u + v − 5e − 9f − 12g − 14h ≤ t + 1. From (4) and (5) we get   t+1 − 2t − 2 < −2b − 3c − 3u − 2v + 15(e + f + g + h) + 2(5e + 9f + 12g + 14h) 2 that is   t+1 − 2t − 2 < −2b − 3c − 3u − 2v + 25e + 33f + 39g + 43h < 125, 2 which implies t2 − 3t − 254 < 0, which is false as soon as t ≥ 18. If e + f + g + h ≤ 2, the same steps give t2 − 3t − 176 < 0 which is false as soon as t ≥ 15. If e = f = g = h = 0, the same steps give t2 − 3t − 4 < 0 which is false as soon as t ≥ 4.  Remark 11. Let Y ⊂ P3 be a zero-dimensional scheme and H a hyperplane of P3 . Fix non negative integers c2 , c3 , c4 , c5 . Denoting by Y ′ the union of the connected components of Y intersecting H, the scheme Y \Y ′ is a general union of c5 5-points, c4 4-points, c3 3-points, and c2 2-points. Moreover, the subscheme Y ′ is supported on general points of H and it is given by a union of points of multiplicity 2, 3, 4, 5 or virtual schemes arising as residual in Lemma 8. In the following basic lemma we show how to apply the Horace differential Lemma 8 in our situation. QUINTUPLE POINTS 9 Lemma 12. Fix a plane H ⊂ P3 . Let Y be a zero-dimensional scheme as in Remark 11, for some integers c2 , c3 , c4 , c5 . If the following condition holds for some positive integer t:   t+2 (6) β := − deg(Y ∩ H) ≥ 0, 2 then it is possible to degenerate Y to a scheme X such that one of the following possibilities is verified:
(I) deg(X ∩ H) = t+2 2
, (II) deg(X ∩ H) < t+2 2 , and all the irreducible components of X are supported on H. This is possible only if c2 + c3 + c4 + c5 < β and c2 + c3 + c4 + c5 ≤ 2. Moreover, if we assume t ≥ 18 in case (I) and t ≥ 15 in case (II), we also have   t+1 (7) deg(ResH (X) ∩ H) ≤ 2 Proof. By specializing some of the connected components of Y to isomorphic schemes supported on points of H we may assume that β ≥ 0 is minimal. Let us denote now by Y ′ the union of the connected components of Y intersecting H. By minimality of β it follows that if c2 > 0 then β < 3, if c2 = 0 and c3 > 0 then β < 6, if c2 = c3 = 0 and c4 > 0 then β < 10, if c2 = c3 = c4 = 0 and c5 > 0, then β < 15. If c2 = c3 = c4 = c5 = 0 and β > 0, we are obviously in case (II). We degenerate now Y to a scheme X described as follows. The scheme X contains all the connected components of Y ′ . Write β = 10e + 6f + 3g + h for a unique quadruple of non-negative integers (e, f, g, h) in the following list: (0, 0, 0, 0), (0, 0, 0, 1), (0, 0, 0, 2), (0, 0, 1, 0), (0, 0, 1, 1), (0, 0, 1, 2), (0, 1, 0, 0), (0, 1, 0, 1), (0, 1, 0, 2), (0, 1, 1, 0), (1, 0, 0, 0), (1, 0, 0, 1), (1, 0, 0, 2), (1, 0, 1, 0), (1, 0, 1, 1) (i.e. in the list of Lemma 10). If c2 > 0, then e = f = g = 0 and h ≤ 2. If c2 = 0 and c3 > 0, then e = f = 0, g ≤ 1 and h ≤ 2. If c2 = c3 = 0 and c4 > 0, then e = 0, f ≤ 1, g ≤ 1, h ≤ 2 and h = 0 if f = g = 1. Consider first the case c2 > 0 and recall that in this case e = f = g = 0 and h ≤ 2. Assume now c2 ≥ h. Take as X a general union of Y ′ , c5 5-points, c4 4-points, c3 3-points, (c2 − h) 2-points, h virtual schemes obtained by applying Lemma 8 at h general points of H with respect to the sequence (1, 3). Clearly we
have deg(X ∩H) = t+2 . Let us see now how to specialize Y to X in the remaining 2 cases with c2 > 0. If c2 = 1 < h and c3 + c4 + c5 ≥ 1, then in the previous step we apply Lemma 8 using the unique 2-point and one 3-point or 4-point or 5-point with respect to the sequence (1, 6, 3) or (1, 10, 6, 3) or (1, 15, 10, 6, 3) (recall that we assumed ci > 0 for at least one i ∈ {3, 4, 5}) and we conclude in the same way. If c2 = 1 < h and c3 = c4 = c5 = 0, then we apply Lemma 8 to the unique double point with respect to the sequence (1, 3), and we are in case (II). Here and in all later instances of case (II) it is straightforward to check that the inequalities c2 + c3 + c4 + c5 < β and c2 + c3 + c4 + c5 ≤ 2 are verified. Assume now c2 = 0 and c3 > 0. Recall that e = f = 0, g ≤ 1 and h ≤ 2. If c3 ≥ g + h we take as X a general union of Y ′ , c5 5-points, c4 4-points, c3 − g − h 3points, g virtual schemes obtained applying Lemma 8 at f general points of H with respect to the sequence (3, 6, 1) and g virtual schemes obtained applying Lemma 8 at g general points of H with respect to the sequence (1, 6, 3). If 0 < c3 < g + h and 10 E. BALLICO, M. C. BRAMBILLA, F. CARUSO, AND M. SALA c4 + c5 ≥ g + h − c3, then in the previous step we apply Lemma 8 using c3 3-points, and (f + g − c3 ) 4-points or 5-points, with respect to the sequences (3, 10, 6, 1) or (1, 10, 6, 3) or (3, 6,
10, 15, 1) or (1, 15, 10, 6, 3). In all these cases we clearly have deg(X ∩ H) = t+2 2 . If c2 = 0, 0 < c3 < g + h and c4 + c5 < g + h − c3 , then we have either c3 = 1 and c4 + c5 ≤ 1, or c3 = 2, g = 1, h = 2 and c4 = c5 = 0. In both cases β > c3 + c4 + c5 . In this cases we can specialize all the components on H, possibly applying Lemma 8 and we are in case (II). Now, assume that c2 = c3 = 0 and c4 > 0. Hence e = 0, f ≤ 1, g ≤ 1, h ≤ 2 and h = 0 if f = g = 1. If c4 + c5 ≥ f + g + h, then we take as X a general union of Y ′ , (c4 + c5 − f − g − h) 4-points or 5-points, f virtual schemes obtained applying Lemma 8 at f general points of H with respect to the sequence (6, 10, 3, 1) or (6, 15, 10, 3, 1), g virtual schemes obtained applying Lemma 8 at g general points of H with respect to the sequence (3, 10, 6, 1) or (3, 15, 10, 6, 1) and h virtual schemes obtained applying Lemma 8 at h general points of H with respect to
the sequence (1, 10, 6, 3) or (1, 15, 10, 6, 3). Thus we have again deg(X ∩H) = t+2 2 . If c2 = c3 = 0 and 0 < c4 + c5 < f + g + h, then we are again in case (II), because we can specialize all the 4-points and 5-points on H (possibly applying Lemma 8), since c4 + c5 ≤ f + g + h ≤ 3 and β = 6f + 3g + h; in this case we may also assume that if f 6= 0 (i.e. f = 1), then either one of the 4-points is specialized with respect to the sequence (6, 10, 3, 1). Finally assume that c2 = c3 = c4 = 0 and c5 > 0. If c5 ≥ e + f + g + h we apply Lemma 8 at e general points of H with respect to the sequence (10, 15, 6, 3, 1), at f general points of H with respect to the sequence (6, 15, 10, 3, 1), at g general points of H with respect to the sequence (3, 15, 10, 6, 1) and at h general points of H with respect to the sequence (1, 15, 10, 6, 3). In this way we arrive to case (I). If e + f + g + h > c5 , then we start applying again Lemma 8 as in the previous stop, but we have to stop at some point and we land in case (II). Finally, we note that the property (7) follows immediately from the construction above and from Lemma 10.  In order to prove the good postulation of schemes in P3 by applying induction, we need to know the good postulation of schemes in P2 . In the next remark we point out the related results that we need. Remark 13. When the general union has multiplicity up to 4 and n = 2, then we can use some results by Mignon (see [18, Theorem 1]). In particular we know that a general fat point scheme in P2 of multiplicity 1 ≤ m ≤ 4 has good postulation in degree d ≥ 3m. Interestingly, this result is valid for any characteristic of the ground field K (for a discussion about char(K) see Section 7). For multiplicities up to 7 and when char(K) = 0, we can use some results by Yang (see [21, Theorem 1 and Lemma 7]), which imply that a general fat point scheme in P2 of multiplicity m ≤ 7 has good postulation in degree d ≥ 3m. The case with no quintuple points has already been solved, as explained below. Remark 14. When the general union Y has multiplicity up to 4 and n = 3, we know that Y must have good postulation in any degree d ≥ 9, thanks to [4] and [10]. There is no self-contained theoretical proof for this, but we have a theoretical proof for d ≥ 41 in [4], along with a computer check up to d = 13, and the missing computations can be found in [10]. QUINTUPLE POINTS 11 2.1. Summary of our computational results. We list the results from Section 5 that we need in the following sections. Lemma 15. The following linear systems are non-special and have virtual dimension −1: (1) L3 (3; 25 ), (2) L3 (9; 4a , 3b ) with 2a + b = 22, (3) L3 (9; 54 , 44 ) (4) L3 (12; 5a , 4b , 3c ) with 7a + 4b + 2c = 91. Theorem 16. Fix non-negative integers d, w, x, y, z such that 11 ≤ d ≤ 21 and
0 ≤ z ≤ 4. Let N = d+3 . Let Y ⊂ P3 be a general union of w 5-points, x 3 4-points, y 3-points and z 2-points such that N − 3 ≤ 35w + 20x + 10y + 4z ≤ N + ∆ , where ∆ is as in Lemma 5. Then Y has good postulation. Theorem 17. Fix non-negative integers d, q, w, x, y, z such that: • 22 ≤ d ≤ 37, • 0 ≤ z ≤ 4, • 0 ≤ 2x + y ≤ 21, • 0 ≤ w ≤ 3 or 0 ≤ x ≤ 3.
3 Let N = d+3 3 . Let Y ⊂ P be a general union of w 5-points, x 4-points, y 3-points and z 2-points such that N − 3 ≤ 220q + 35w + 20x + 10y + 4z ≤ N + ∆ , where ∆ is as in Lemma 5. Then Y has good postulation. Theorem 18. Fix non-negative integers d, r, w, x, y, z such that: • 38 ≤ d ≤ 52, • 0 ≤ z ≤ 4, • 0 ≤ 2x + y ≤ 41, • 0 ≤ w ≤ 12.
3 Let N = d+3 3 . Let Y ⊂ P be a general union of w 5-points, x 4-points, y 3-points and z 2-points such that N − 3 ≤ 455r + 35w + 20x + 10y + 4z ≤ N + ∆ , where ∆ is as in Lemma 5. Then Y has good postulation. 12 E. BALLICO, M. C. BRAMBILLA, F. CARUSO, AND M. SALA 3. The proof of Theorem 1 for high degrees This section is devoted to the proof of Theorem 1 for high degrees, that is for d ≥ 53. Throughout the section we fix a hyperplane H ⊂ P3 . We recall that our ground field K has characteristic zero. In the different steps of the proof we will work with zero-dimensional schemes that are slightly more general than a union of fat points. In particular, we will say that a zero-dimensional scheme Y is of type (⋆) if its irreducible components are of the following type: - m-points with 2 ≤ m ≤ 5, supported on general points of P3 , - m-points with 1 ≤ m ≤ 5, or virtual schemes arising as residual in the list of Lemma 8, supported on general points of H. Given a scheme Y of type (⋆) satisfying (6) for some integer t, we will say that Y is of type (I,t) if, when we apply Lemma 12 to Y , we are in case (I). Otherwise we say that Y is of type (II,t). We fix now (and we will use throughout this section) the following notation, for any integer t: given a scheme Yt of type (⋆) and satisfying (6) for t, we will denote by Xt the specialization described in Lemma 12. We write the residual ResH (Xt ) = Yt−1 ∪ Zt−1 , where Yt−1 is the union of all unreduced components of ResH (Xt ) and Zt−1 = ResH (Xt ) \ Yt−1 . Clearly Zt−1 is the union of finitely many simple points of H. Thus, at each step t 7→ t − 1 we will have Yt 7→ Xt 7→ ResH (Xt ) = Yt−1 ∪ Zt−1 . For any integer t, we set zt := |Zt |, αt := deg(Yt ) = deg(Xt ), and     t+2 δt := max 0, − deg(Yt−1 ∪ Zt−1 ) . 3 We fix the following statements: - A(t) = {Yt has good postulation in degree t}, - B(t) = {ResH (Xt ) has good postulation in degree t − 1}, - C(t) = {h0 (P3 , IResH (Yt−1 ) (t − 2)) ≤ δt }. Claim 19. Fix t ≥ 16. If Yt is a zero-dimensional scheme of type (II,t), then it has good postulation, i.e. A(t) is true. Moreover if t ≥ 17, also B(t) is true. Proof. Since Yt is of type (II,t), when we apply Lemma 12 to Yt , we obtain a specialization Xt such that all
its irreducible components are supported on H and such that deg(Xt ∩ H) ≤ t+2 2 . We prove now the vanishing h1 (P3 , IYt (t)) = 0. By semicontinuity, it is enough to prove the vanishing h1 (P3 , IXt (t)) = 0. Notice that by taking the residual of Xt with respect to H for at most
five times we get at the end the empty set. Since deg(Xt ∩ H) ≤ t+2 and t ≥ 15, by Remark 13 it follows the vanishing 2 1 2 h (P , IXt ∩H (t)) = 0. Let Rt−1 denote the residual ResH (Xt ) and recall that any component of Rt−1 is supported on H. We check now that h1 (P3 , IRt−1 (t−1)) = 0. In order to do this we take again the trace and
the residual with respect to H. By (7) we know that deg(ResH (Xt ) ∩ H) ≤ t+1 then again by Remark 13, since 2 t − 1 ≥ 15, we have h1 (P2 , IRt−1 ∩H (t − 1)) = 0. We repeat this step taking Rt−2
:= ResH (Rt−1 ) and noting that the trace Rt−2 ∩ H has degree less or equal than 2t , by Lemma 10. Moreover this time the scheme Rt−2 ∩ H cannot contain quintuple points, in fact it is a general union of quartuple, QUINTUPLE POINTS 13 triple, double and simple points. Hence by Remark 13 we have h1 (P2 , IRt−2 ∩H (t − 2)) = 0, since t − 2 ≥ 12. We repeat once again the same step and we obtain Rt−3 := ResH (Rt−2 ). Now the trace Rt−3 ∩ H contains only triple, double or simple points and so we have again the vanishing h1 (P2 , IRt−3 ∩H (t − 3)) = 0, by Remark 13, since t − 3 ≥ 9. Set Rt−4
:= ResH (Rt−3 ). The scheme Rt−4 ∩ H is reduced and formed by less than t−2 general points of H. Hence h1 (P2 , IRt−4 ∩H (t − 4)) = 0. Notice that this 2 time the residual ResH (Rt−4 ) must be empty and so, since IResH (Rt−4 ) = OP3 , we obviously have h1 (P3 , IResH (Rt−4 ) (t − 5)) = 0. Hence thanks to Lemma 7 we obtain h1 (P3 , IYt (t)) = 0. We also know that (8)             t+2 t+1 t t−1 t−2 t+3 deg(Yt ) = deg(Xt ) ≤ + + + + ≤ 2 2 2 2 2 3
where the second inequality is equivalent to t−2 ≥ 0, which is true if t ≥ 4. Hence 3 Yt has good postulation, that is, A(t) is true. It is easy to see that also the scheme Res(Xt ) must be of type (II,t − 1). Hence B(t) follows from the first part of the proof.  Claim 20. Fix t ≥ 15. If Yt is a zero-dimensional scheme of type (I,t), then A(t) is true if B(t) is true. Proof. Since Yt is of type (I,t), we can apply
Lemma 12 and we obtain a specialization Xt such that deg(Xt ∩ H) = t+2 2 . Thus, since t ≥ 15, by Remark 13 it follows h0 (H, IXt ∩H (t)) = h1 (H, IXt ∩H (t)) = 0. Then, thanks to Lemma 7, it follows, for i = 0, 1, hi (P3 , IXt (t)) = hi (P3 , IResH (Xt ) (t − 1)). Thus in order to prove that the scheme Xt has good postulation in degree t, it is sufficient to check the good postulation of ResH (Xt ) in degree t − 1.  Claim 21. If A(t − 1) and C(t) are true, then B(t) is true. Proof. Recall that we write ResH (Xt ) = Yt−1 ∪Zt−1 ,where Zt−1 is a union of simple points supported on H. By [4, Lemma 7], to check that the scheme ResH (Xt ) has good postulation in degree t − 1 (i.e. B(t)), it is sufficient to check the good postulation of Yt−1 in degree t − 1 (i.e. A(t − 1)) and to prove that C(t) is true.  Claim 22. If Yt is of type (I,t), then B(t − 1) implies C(t). Proof. The statement C(t) is true if h0 (P3 , IResH (Yt−1 ) (t − 2)) ≤ δt . Note that since
deg(Xt ∩ H) = t+2 2 , we have   t+2 deg(ResH (Xt )) = deg(Yt−1 ∪ Zt−1 ) = αt−1 + zt−1 = αt − , 2 and thus it follows         t+3 t+2 − αt . δt := max 0, − αt−1 − zt−1 = max 0, 3 3 14 E. BALLICO, M. C. BRAMBILLA, F. CARUSO, AND M. SALA
. Hence, it follows     t+2 t+1 deg(ResH (Yt−1 )) = deg(ResH (ResH (Xt ))) ≥ αt − − 2 2



and then, since t+2 + t+1 = t+3 − t+1 2 2 3 3 , we get       t+1 t+3 t+1 deg(ResH (Yt−1 )) ≥ − + αt ≥ − δt . 3 3 3 Notice that, by (7), we have deg(ResH (Xt ) ∩ H) ≤ t+1 2 So in order to prove C(t) it is enough to prove that ResH (Yt−1 ) has good postulation in degree t − 2.  Now we are in position to prove our main result. Proof of Theorem 1 for d ≥ 53. For all non negative integers d ≥ 53 and w, x, y, z, we set   d+3 ǫ(d, w, x, y, z) := − 35w − 20x − 10y − 4z. 3 We will often write ǫ instead of ǫ(d, w, x, y, z) in any single step of the proofs in which the parameters d, w, x, y, z are fixed. By Lemma 5, in order to prove our statement for all quadruples (w, x, y, z) it is sufficient to check it for all quadruples (w, x, y, z) such that −13 ≤ ǫ(d, w, x, y, z) ≤ 3. We fix any such quadruple and we consider a general union Y of w 5-points, x 4-points, y 3-points and z 2-points. Notice that       1 d+3 3 d+3 1 −3 ≥ − , (9) w+x+y+z ≥ 35 35 35 3 3 l  m
d+3 1 i.e. the scheme Y has at least 35 − 3 connected components. 3 The proof is by induction, based on Lemma 7, and it requires different steps. Set Yd = Y and
fix a hyperplane H ⊂ P3 . We can assume by generality that deg(Yd ∩ H) ≤ d+2 2 , hence we can apply Lemma 12, thus specializing the scheme Yd to a scheme Xd . If Yd is of type (II,d), then we conclude by Claim 19, since d ≥ 16. Hence we can assume that Yd is of type (I,d), and so, since d ≥ 15, by Claim 20 it is enough to check that the scheme ResH (Xd ) has good postulation in degree d − 1. Now we write ResH (Xd ) = Yd−1 ∪ Zd−1 , where Yd−1 is the union of all unreduced components of ResH (Xd ) and Zd−1 = ResH (Xd ) \ Yd−1 . By Claim 21, it is enough to prove that A(d − 1) and C(d) are true. Notice
that, since d ≥ 18, we get (7), i.e. deg(Yd−1 ∩ H) ≤ deg(ResH (Xd ) ∩ H) ≤ d+1 2 . Hence Yd−1 satisfies condition (6) in degree d − 1, then we can apply again Lemma 12. We have now two alternatives: either Yd−1 is of type (I,d−1), or of type (II,d−1). In both cases, we note that by Claim 22 the statement C(d) follows from B(d − 1), since Yd is of type (I,d). Now assume that Yd−1 is of type (II,d − 1). Then by Claim 19, since d − 1 ≥ 17 we know that B(d − 1) and A(d − 1) are true and this concludes the proof. It remains to consider the case Yd−1 of type (I,d − 1). We apply again Claim 21 and we go on iterating the same steps. Now we have two cases: either in a finite number v of steps the procedure described above gives us a scheme Xd−v of type (II,d − v), for a degree d − v ≥ 18, or the procedure goes on until we get X18 , a scheme of type (I,18). QUINTUPLE POINTS 15 In the first case, the steps of the procedure above prove that the scheme Xd has good postulation, and the statement is proved. Assume now that
we are in the second case, i.e. X18 is of type (I,18), that is, deg(X18 ∩ H) = 20 2 . Note that, since ǫ ≥ −13, we have     d−1 d−1 X X 21 21 (10) deg(X18 ) = −ǫ− zt ≤ + 13 − zt . 3 3 t=18 t=18 P Now we want to estimate d−1 t=18 zt , which is the number of simple points we have removed in the steps above. Since we started from the scheme Yd , in d − 18 steps we arrived at the scheme X18 in such a way that the case (II) never occurred. Assume that in these d − 18 steps we have applied γ times Lemma 8 with respect to sequences of type (1, 15, 10, 6, 3), (1, 10, 6, 3), (1, 6, 3) or (1, 3). As it is clear looking at the proof of Lemma 12, at each step the number of times we used a sequence giving as a trace a simple point is at most 2, hence we have γ ≤ 2(d − 18). Let u18 denote the number of connected components of X18 . Hence it follows that (11) d−1 X zt ≥ w + x + y + z − 2(d − 18) − u18 . t=18 Now we need to estimate the number u18 . Let us denote by T the union of components of X18 of length 3.
Then any component of the scheme X18 \ T has length at least 4, and deg(T ) ≤ 20 2 since the scheme T is completely contained in the trace X18 ∩ H. So we have   1 1 1 20 1 + (deg(X18 )), u18 ≤ deg(T ) + (deg(X18 ) − deg(T )) ≤ 3 4 12 2 4 and using (11) and (9), we get     d−1 X 3 1 1 d+3 1 20 − − (deg(X18 ))). (12) zt ≥ − 2(d − 18) − 35 35 12 4 3 2 t=18 By using (10) and (12) we get       3 1 20 1 d+3 21 3 + deg(X18 ) ≤ + 2(d − 18) + + 13 − 4 35 35 12 2 3 3
21 and, since d ≥ 53, it is easy to check that deg(X18 ) ≤ 3 . Note that X18 depends implicitely on d, but we use the above inequality to show what happend for d = 53. Of cousre, for higher d’s we can easily show that deg(X18 ) is actually even smaller,
but we do not need it and we content ourselves with the claimed deg(X18 ) ≤ 21 3 Hence we need to prove the vanishing h1 (P3 , IX18 (18)) = 0, and by Claim 12, it is enough to prove that h1 (P3 , IResH (X18 ) (17)) = 0. Now we change the procedure. Denote ResH (X18 ) = R17 .
Since deg(R17 ∩ H) ≤ 19 , specializing some points on H we can degenerate 2 e17 (without applying the Horace differential lemma) the scheme R17 to a scheme R in such a way that one of the following cases happens:
e17 ∩H) ≤ 19 −15 and all the components of R e17 are supported (1a) either deg(R 2 on H,

19 e (1b) or 19 2 − 14 ≤ deg(R17 ∩ H) ≤ 2 . 16 E. BALLICO, M. C. BRAMBILLA, F. CARUSO, AND M. SALA e17 = E17 ∪ F17 , where E17 is supported on H and Denote now, in both cases, R e17 ∩H = E17 ∩H and the residual F17 is supported outside H. Take now the trace R e17 ) = R16 . ResH (R Note that in case (1a) F17 = ∅, while in case (1b)      20 19 deg(R16 ) ≤ deg(X18 ) − − − 14 . 2 2 By Lemma 7, to prove that h1 (P3 , IR17 (17)) = 0, it is enough to prove that h (P2 , IE17 ∩H (17)) = 0 (by Remark 13, since 17 ≥ 15) and h1 (P3 , IR16 (16)) = 0. Now we repeat the same step, that is, we specialize some points on H without e16 in such a way that applying the Horace differential lemma, degenerating R16 to R one of the following cases happens:
e16 ∩H) ≤ 18 −15 and all the components of R e16 are supported (2a) either deg(R 2 on H,

18 e (2b) or 18 2 − 14 ≤ deg(R16 ∩ H) ≤ 2 . e16 = E16 ∪ F16 , where E16 is supported on H and Denote again, in both cases, R F16 is supported outside H. Note that E16 is given by quintuple, quartuple, triple, double and simple points or virtual schemes arised by the application of Lemma 10. In any case taking the residual with respect to H five times we get that the last residual has no components supported on H. e16 ∩ H = E16 ∩ H and the residual ResH (R e16 ) = R15 . Take now the trace R Note that in case (2a) F16 = ∅, while in case (2b)         18 20 19 − 14 . − 14 − deg(R15 ) ≤ deg(X18 ) − − 2 2 2 1 By Lemma 7, to prove h1 (P3 , IR16 (16)) = 0, we only need h1 (P2 , IE16 ∩H (16)) = 0 (which is true by Remark 13, since 16 ≥ 15) and h1 (P3 , IR15 (15)) = 0. Now, without specializing furtherly, we denote R15 = E15 ∪ F15 , where E15 is supported on H and F15 = F16 is supported outside H. Take now the trace e15 ∩ H = E15 ∩ H and the residual ResH (R e15 ) = R14 . R 1 3 By Lemma 7, to prove that h (P , IR15 (15)) = 0, it is enough to prove that h1 (P2 , IE15 ∩H (15)) = 0 (which is true by Remark 13, since 15 ≥ 12 and the trace contains at most quartuple points) and h1 (P3 , IR14 (14)) = 0. We repeat again the same step and we get R14 = E14 ∪ F14 , where E14 is supported on H and F14 = F16 is supported outside H. e14 ∩ H = E14 ∩ H and the residual ResH (R e14 ) = R13 . Take now the trace R By Lemma 7, to prove that h1 (P3 , IR14 (14)) = 0, it is enough to prove that h1 (P2 , IE14 ∩H (14)) = 0 (which is true by Remark 13, since the trace contains at most triple points) and h1 (P3 , IR13 (13)) = 0. We repeat again the same step and we get R13 = E13 ∪ F13 , where E13 is supported on H and F13 = F16 is supported outside H. e13 ∩ H = E13 ∩ H and the residual ResH (R e13 ) = R12 . Take now the trace R 1 3 By Lemma 7, to prove that h (P , IR13 (13)) = 0, it is enough to prove that h1 (P2 , IE13 ∩H (13)) = 0 (which is true by Remark 13, since the trace contains at most double points) and h1 (P3 , IR12 (12)) = 0. Now we take again for the last time the trace and the residual with respect to H. Denote R12 = E12 ∪ F12 , where E12 is supported on H and F12 = F16 is supported QUINTUPLE POINTS 17 outside H. Taking the trace and the residual we have that E12 ∩ H is given by general simple points in H and obviously we have h1 (P2 , IE12 ∩H (12)) = 0. So we need only to show that the residual ResH (E12 ∪ F12 ) = R11 satisfies h1 (P3 , IR11 (11)) = 0. Note that the residual does not have components supported on H. More precisely, R11 = F12 = F16 , that is, the residual is a general collection of double, triple, quartuple and quintuple points. But Theorem 16 ensures that any general collection of double, triple, quartuple and quintuple points has good postulation in degree 11. So in order to conclude the proof of the theorem it is enough to prove the following inequality:   14 (13) deg(R11 ) = deg(F16 ) ≤ . 3 Let us check this condition in any of the previous cases: in cases (1a) and (2a) we have F16 = ∅ and so condition (13) is obviously satisfied. It remains to prove (13) in case (2b), where         20 19 18 deg(R11 ) ≤ deg(R15 ) ≤ deg(X18 ) − − − 14 − − 14 . 2 2 2 By (10) and (12) we have         3 1 1 20 1 d+3 21 − − (deg(X18 ))) − 2(d − 18) − +13− deg(X18 ) ≤ 35 3 35 12 2 4 3 from which we obtain       4 1 d+3 3 21 1 20 deg(X18 ) ≤ + 13 − + + 2(d − 18) + 3 35 35 12 2 3 3 and so         20 19 18 deg(R15 ) ≤ deg(X18 ) − − − 14 − − 14 . 2 2 2 It is easy to check that, for any d ≥ 53 the inequality            1 d+3 3 21 20 1 20 19 4 + 13 − + − − − 14 + 2(d − 18) + 3 35 35 12 2 3 3 2 2      18 14 − − 14 ≤ 2 3 is verified, and this implies (13) and completes the proof.  18 E. BALLICO, M. C. BRAMBILLA, F. CARUSO, AND M. SALA 4. The proof of Theorem 1 for low degrees In this section we discuss Theorem 1 in the remaining cases, that is, when the degree d satisfies 11 ≤ d ≤ 52. In these cases the proof is based on computer calculations, which are described explicitly in Section 5. Although in principle it is possible to go through all cases in Lemma 5 for 11 ≤ d ≤ 52, this is impractical with nowadays computers. In order to shorten the computational time we need some other auxiliary theoretical results, that we develop in this section. First we prove Theorem 1 for degrees ≥ 38 in the special case when we have few quintuple points
1 d+3 (more precisely when 35w ≤ 12 ). Then we will present how to apply a re3 sult by Dumnicki in order to greatly reduce the cases to be tested by our computers. The proof of the following proposition is a modification of the argument in the previous section, where we proved Theorem 1 for d ≥ 53. Proposition 23. Fix non-negative integers d ≥ 38, w, x, y, z such that 35w ≤
1 d+3 and 12 3     d+3 d+3 − 3 ≤ 35w + 20x + 10y + 4z ≤ + 13. 3 3 Let Y ⊂ P3 be a general union of w 5-points, x 4-points, y 3-points and z 2-points. Then Y has good postulation in degree d. Proof. We follows the same procedure as in the main proof of Section 3. The first difference is that every time we apply Lemma 12, we specialize on the plane H as many quintuple points as possible. So starting with Yd = Y , we obtain in a finite number of steps d − d0 a scheme Xd0 which does not contain quintuple points. We prove
that in particular d0 ≥ 20. 1 d+3 Indeed sinceby assumption we have 35w ≤ and d ≥ 20, it easily follows 12 3

 d+3 22 1 − 3 . that w ≤ 35 3 Thus we have a general union Xd0 of quartuple, triple and double points, and of virtual schemes of the type listed in the table of Lemma 8, arised by the application of Lemma 12. If Xd0 is of type (II,d0 ), then we conclude, as in the previous proof, that it has good postulation and this implies that Y has good postulation. Let us assume that Xd0 is of type (I,d0 ). Applying again Lemma 12 we can go on with our usual argument and we will obtain or a scheme of type (II,e), for some e ≥ 18, which concludes the proof, or a scheme X18 of type (I,18) and without quintuple points. At this point we can apply the same argument used in the proof of [4, Theorem 1], regarding union of quartuple, triple and double points and virtual schemes of the type listed in [4, Lemma 4]. In particular we apply [4, Lemma 8] until we get a scheme Xe of type (II,e) in degree e ≥ 13. In this case we conclude, as in [4], that our scheme has good postulation. Now it remains to consider the case when we get a scheme X13 of type (I,13). Notice that in this case we want to prove that h1 (P3 , IX13 (13)) = 0. QUINTUPLE POINTS Indeed let us prove that deg(X13 ) ≤
1 d+3 12 3 ,
  3 1 11 d + 3 − · 20 12 20 3 d+3 3 − 35w − 20x − 10y − 4z ≥ −13 we have:     d−1 d−1 X X 16 16 deg(X13 ) = −ǫ− zt ≤ + 13 − zt , 3 3 t=13 t=13 and setting ǫ = (15)
. First of all, note that, since 35w ≤ 16 3 w+x+y+z ≥x+y+z ≥ (14) 19 where zt denotes, as in Section 3, the number of simple points we have removed at the (d − t)-th step. As in (11) we have d−1 X zt ≥ w + x + y + z − 2(d − 13) − u13 , t=13 where u13 is the number of connected component of X13 . Since X13 does not contain simple points we have u13 ≤ 31 deg(X13 ) and so by (14) we get   d−1 X 11 d + 3 1 3 (16) zt ≥ − 2(d − 13) − (deg(X13 )) − 240 20 3 3 t=13 and by (15) we get      3 11 d + 3 3 16 + 13 − + + 2(d − 13) 2 240 20 3 3 But now it is easy to check that        11 d + 3 3 16 3 16 + 13 − + + 2(d − 13) ≤ 2 240 20 3 3 3 (17) deg(X13 ) ≤ as soon as d ≥ 30. Then it is enough to prove that h1 (P3 , IX13 (13)) = 0. Now we apply the residual without specializing any further components on H. In other words we take Y12 := ResH (X13 ), Y11 := Res H
H (Y12 ), Y10 := Res
(Y11 ) 13 and Y9 := ResH (Y10 ). Notice that deg(Y12 ∩ H) ≤ 14 , deg(Y ∩ H) ≤ 11 2 2 , and
12 1 2 deg(Y10 ∩ H) ≤ 2 . So by Remark 13 all the vanishings h (P , IY12 ∩H (12)) = 0, h1 (P2 , IY11 ∩H (11)) = 0 and h1 (P2 , IY10 ∩H (10)) = 0 are satisfied. Hence by Lemma 7, it is sufficient to prove h1 (P3 , IY9 (9)) = 0. Recall that for any integer t ≥ 9 a general union of quadruple, triple and double points has good postulation in degree t by [4, 10].
Thus it is sufficient to prove that deg(Y9 ) ≤ 12 3 . Indeed obviously we have deg(Y9 ) ≤ deg(X13 ). It is easy to check that        11 d + 3 3 16 12 3 + 13 − + + 2(d − 13) ≤ 2 240 20 3 3 3
12 for any d ≥ 38. Hence by (17) we have deg(Y9 ) ≤ 3 and this concludes our proof.  The crucial tool which allow us to perform our computation in a reasonable time is the following special case of [9, Theorem 1]. Theorem 24 (Dumnicki). Let d, k, m1 , . . . , ms , ms+1 . . . , mr ∈ N. If • L1 = L3 (k; m1 , . . . , ms ) is non-special; 20 E. BALLICO, M. C. BRAMBILLA, F. CARUSO, AND M. SALA • L2 = L3 (d; ms+1 , . . . , mr , k + 1) is non-special; • vdimL1 = −1 then the system L = L3 (d; m1 , . . . , mr ) is non-special. Remark 25. To obtain Theorem 24 we have applied [9, Theorem 1] to the case n = 3 and vdim(L1 ) = −1, since the latter clearly guarantees (vdimL1 + 1)(vdimL2 + 1) ≥ 0. Although this is apparently very restrictive, in practice it is very difficult to find different applications which perform efficiently. The next three lemmas explain how to use Theorem 24 in order to reduce the computations.
Lemma 26. Fix a positive integer d and let N = d+3 3 . For any quadruple of non-negative integers (w, x, y, z), let Y (w, x, y, z) ⊂ P3 denote a general union of w 5-points, x 4-points, y 3-points and z 2-points. If Y (w, x, y, z) has good postulation in degree d for any quadruple (w, x, y, z) such that 0 ≤ z ≤ 4, N − 3 ≤ 35w + 20x + 10y + 4z ≤ N + ∆, where ∆ is defined as in (1), then any general quintuple fat point scheme has good postulation in degree d. Proof. Let Y = Y (w, x, y, z) be a general quintuple fat point scheme. Recall that by Lemma 5 it is enough to prove the good postulation of Y when N − 3 ≤ 35w + 20x + 10y + 4z ≤ N + ∆. Now assume that z ≥ 5. By Lemma 15 we know that L3 (3; 25 ) is non-special and vdim(L3 (3; 25 )) = −1. Then by Theorem 24 in order to prove that Y has good postulation in degree d, it is enough to prove that Y (w, x + 1, y, z − 5) has good postulation. Repeating this step, we reduce to the case when z ≤ 4, and this proves our lemma. 
Lemma 27. Fix a positive integer d and let N = d+3 3 . Given non-negative integers q, w, x, y, z, let Y (w, x, y, z) ⊂ P3 denote a general union of w 5-points, x 4-points, y 3-points and z 2-points and let Y ′ (q, w, x, y, z) denote the union of q general 10-fat points with Y (w, x, y, z). If Y ′ (q, w, x, y, z) has good postulation in degree d for any quintuple (q, w, x, y, z) such that 0 ≤ z ≤ 4, 0 ≤ 2x + y ≤ 21, 0 ≤ w ≤ 3 or 0 ≤ x ≤ 3, N − 3 ≤ 220q + 35w + 20x + 10y + 4z ≤ N + ∆, where ∆ is defined as in (1), then any general quintuple fat point scheme has good postulation in degree d. Proof. Let Y = Y (w, x, y, z) be a general quintuple fat point scheme. As in the proof of Lemma 26 we can assume N − 3 ≤ 35w + 20x + 10y + 4z ≤ N + ∆ and z ≤ 4. Now assume that 2x + y ≥ 22. Then there exist two integers a, b such that a ≤ x and b ≤ y and 2a + b = 22. By Lemma 15 we know that the linear system L3 (9; 4a , 3b ) is non-special and with virtual dimension −1. So by Theorem 24 in order to prove that Y has good postulation in degree d, it is enough to prove that Y ′ (1, w, x − a, y − b, z) has good postulation. Repeating this step, we reduce to check all the general unions Y ′ (q, w, x′ , y ′ , z) such that N − 3 ≤ 220q + 35w + 20x + 10y + 4z ≤ N + ∆ and 2x′ + y ′ ≤ 21. QUINTUPLE POINTS 21 Now assume that w ≥ 4 and x′ ≥ 4. By Lemma 15 we know that L3 (9; 54 , 44 ) is non-special and with virtual dimension −1. Thus by Theorem 24 it is enough to prove that Y ′ (q + 1, w − 4, x′ − 4, y ′ , z) has good postulation. Repeating this step, we complete the proof of the lemma. 
Lemma 28. Fix an integer d ≥ 38 and let N = d+3 3 . Given non-negative integers r, w, x, y, z, let Y (w, x, y, z) ⊂ P3 denote a general union of w 5-points, x 4-points, y 3-points and z 2-points and let Y ′′ (r, w, x, y, z) denote the union of r general 13fat points with Y (w, x, y, z). If Y ′′ (r, w, x, y, z) has good postulation in degree d for any quintuple (r, w, x, y, z) such that 0 ≤ z ≤ 4, 0 ≤ w ≤ 12, 0 ≤ 2x + y ≤ 41, N − 3 ≤ 455r + 35w + 20x + 10y + 4z ≤ N + ∆, where ∆ is defined as in (1), then any general quintuple fat point scheme has good postulation in degree d. Proof. Let Y = Y (w, x, y, z) be a general quintuple fat point scheme. As in the proof of Lemma 26 we can assume N − 3 ≤ 35w + 20x + 10y + 4z ≤ N + ∆ and z ≤ 4.
1 d+3 Let α = ⌊ 2x+y 42 ⌋. Now if w ≤ α−1, then by Lemma 9 we also have 35w ≤ 12 3 and we can apply Proposition 23 which says that Y has good postulation. Assume now that w P ≥ α. For 1 ≤ i ≤ α, let ai , bi be such that 2ai + bi = 42 for P α all i, α a ≤ x and i i=1 i=1 bi ≤ y. Note that by Lemma 15 all the linear systems ai bi L3 (12; 5, 4 , 3 ) are non-special and with virtual dimension −1, for 1 ≤ i ≤ α. Then in order to prove that Y has good postulation in degree P d, we apply α times Theorem 24 and we reduce to prove that Y ′′ (α, w − α, x − ai , y − bi , z). So we have to check all the unions of the form Y ′′ (r, w′ , x′ , y ′ , z), where 0 ≤ 2x + y ≤ 41 and N − 3 ≤ 455r + 35w′ + 20x′ + 10y ′ + 4z ≤ N + ∆. Now assume that w′ ≥ 13 and recall that by Lemma 15 the linear system L3 (12; 513 ) is non-special and with virtual dimension −1. Then applying Theorem 24 we reduce to the case when the number of quintuple points is less or equal then 12, and this completes the proof.  We are now in position to complete the proof of Theorem 1. Proof of Theorem 1 for 11 ≤ d ≤ 52.
Let d satisfy 11 ≤ d ≤ 21 and let N = d+3 3 . Lemma 26 says that to prove the good postulation of any general union it is enough to check all the general unions with 0 ≤ z ≤ 4, and N − 3 ≤ 35w + 20x + 10y + 4z ≤ N + ∆, where ∆ is defined as in (1). This is precisely Theorem 16. Now assume that 22 ≤ d ≤ 37. By Lemma 27 it is enough to prove that a general union of q 10-points, w quintuple points, x quartuple points, y triple points and z double points has good postulation, when 0 ≤ z ≤ 4, 0 ≤ 2x + y ≤ 21, 0 ≤ w ≤ 3 or 0 ≤ x ≤ 3 and N − 3 ≤ 220q + 35w + 20x + 10y + 4z ≤ N + ∆. This is Theorem 17. Finally if 38 ≤ d ≤ 52, Lemma 28 proves that it is enough to check all the general unions of r 13-points, w quintuple points, x quartuple points, y triple points and z double points have good postulation, when 0 ≤ z ≤ 4, 0 ≤ w ≤ 12, 0 ≤ 2x + y ≤ 41 and N − 3 ≤ 455r + 35w + 20x + 10y + 4z ≤ N + ∆. This is precisely Theorem 18. This concludes the proof of Theorem 1.  22 E. BALLICO, M. C. BRAMBILLA, F. CARUSO, AND M. SALA 5. A computational proof for the remaining cases In this section we show how several computer calculations allow to prove Lemma 15, Theorem 16, 17, 18. The core of our computation is a programme exact case.magma, that can be found at http://www.science.unitn.it/~sala/fat_points We can idealize the operations performed by exact case.magma as in the following pseudo-code description of a routine called exact. Exact Input (w,x,y,d). N:=Binomial(d+3,3); MonomialMatrices(MList); L:=35*w+20*x+10*y+4*z; // Length // We create the matrix and compute its rank BigM:=EvaluationMatrix(MList,q,w,x,y,z); r:=Rank(BigM); // We check the speciality if ((L lt N) and (r ne L)) then WriteToFile([q,w,x,y,z]); end if; if ((L ge N) and (r ne N)) then WriteToFile([q,w,x,y,z]); end if; WriteToFile(certificate); The first function MonomialMatrices creates a list of matrices MList = {M2 , M3 , M4 , M5 } with monomials entries, where for all matrices the columns correspond to all degreed monomials in four variables, and the rows of Mh correspond to the conditions (partial derivatives) of points with multiplicity h. This list is passed to function EvaluationMatrix alongside with the number of points of given multiplicities. The function EvaluationMatrix creates a set of corresponding random points with coordinates in the finite field Fp . The matrices in MList are evaluated at this set. The output matrix is stored into BigM, whose rank is computed immediately afterwards. Depending on the rank and on the length, if the point configuration is special then a line is written, otherwise no extra output is needed (see later for a discussion on the certificate). Several comments on the above algorithm and its implementation are in order: • The algorithm as described is non-deterministic because it uses random points; we have limited ourselves to use pseudorandom sequences and so we need to choose a seed (and a step) whenever we launch an instance of the procedure, making the algorithm deterministic. In practice, we use QUINTUPLE POINTS 23 the in-built MAGMA pseudo-random generator: Magma contains an implementation of the Monster random number generator by G. Marsaglia ([16]) combined with the MD5 hash function. The period of this generator is 229430 − 227382 and passes all tests in the Diehard test suite ([17]). • The bottle-neck of the algorithm is the rank computation. Although in principle it is possible to check the matrix rank over Q, in practice it is much more efficient to perform these computations over a finite field Fp , with p a prime. This is lecit thanks to Remark 29. The smaller p is, the faster the rank computation is (and the smaller the memory requirement); however, a smaller prime is more likely to trigger a wrong rank (failure), because of the larger number of triggered linear relations; therefore, it is important to find a prime which is both small enough to use a reasonable memory amount and large enough to avoid failures, if possible. It turns out that p = 31991 works well up to the degrees that we needed. Its size is also very close to 215 , and so the computer will allocate exactly 2 bytes to represent it, without losing an overhead. • The rank computation itself is performed by the internal MAGMA rank routine for dense matrices over finite fields. By using several optimization techniques, it can compute the ranks also for large matrices in a reasonable time. We did some tests and MAGMA’s rank routine not only outperforms by far any other software package we tried, but it also competes with ad-hoc compiled programmed using specialized libraries, such as FFLAS-FFPACK ([11]) or M4RI ([14]), although the matrices are not so large as to take advantage of sophisticated algorithms such as Strassen’s ([19]) or Winograd’s ([20]). • The algorithm writes a digital certificate, i.e. a file containing vital information enabling a third party to check the correctness of the output. Our certificates vary slightly depending on the cases examined, but in each we need: the MAGMA’s version, the input variables, the pair seed/step, the prime, the total computation time and a list of failures (if any). Anyone reading a certificate is able to run the corresponding procedure instance and verify the output (assuming that our same pseudorandom sequence is utilized). Remark 29. Let d ≥ 11 be an integer and p be a prime. As usual, let K be any field with characteristic zero. Given a quadruple of integers (w, x, y, z), the computer finds (in absence of failures) a union Y (w, x, y, z) ⊂ P3 (Fp ) that is not defective in degree d. By semicontinuity, this proves that a general union of w 5-points, x 4-points, y 3-points and z 2-points defined over Fp is not defective in degree d. By semicontinuity this is true for a general union Y (w, x, y, z) defined over Fp . By semicontinuity this is also true for a general union Y (w, x, y, z) defined first over Q and then over K. Thanks to Lemma 2 this holds also over K. The first cases that we checked are the small-degree cases in Lemma 15. The programme and the digital certificates can be found at http://www.science.unitn.it/~sala/fat_points/small_cases Although cases a) and b) of Lemma 15 were already known in [10], we redid also them for completeness and check. 24 E. BALLICO, M. C. BRAMBILLA, F. CARUSO, AND M. SALA To check the cases in Theorem 16 we prepared a slightly more complex programme, fat points brutal.magma. We obviously reuse exact but we have to take into consideration the ∆ values from Lemma 5. A pseudo-code description goes as follows. Check of cases 11-21 Input: d. N:=Binomial(d+3,3); // We determine the maximum number of points z1:=4;y1:=Ceiling(N/10);x1:=Ceiling(N/20);w1:=Ceiling(N/35); // We set the maximum value of _D, but since the computations are fast // we leave it except for z>0 _D:=13; for z in [0..z1] do if (z gt 0) then _D:=1; end if; for y in [0..y1] do for x in [0..x1] do for w in [1..w1] do // we start from w=1, because w=0 is already in [10] L:=35*w+20*x+10*y+4*z; // Length if ((L gt N-4) and (L lt N+_D+1)) then exact(w,x,y,d); end if; end for; end for; end for; end for; The programme and the digital certificates can be found at http://www.science.unitn.it/~sala/fat_points/11-21/ We report the timings in the following table. Table 1. Timings in seconds for d = 11 . . . 21 from Theorem 17 d 11 54 12 137 13 309 14 683 15 1449 16 2879 17 5736 18 11016 19 19857 20 35707 21 61171 We proved Theorem 17 similarly, using our programme fat points 10p.magma. We do not give a pseudo-code, since now it is quite obvious how we proceed. We note only two key differences. First of all, we used fully the advantage offered by the tight determination of ∆. Second, we needed also 10-degree points, but this offered no difficulty, since a slight modification of exact can handle them easily. The programme and the digital certificates can be found at http://www.science.unitn.it/~sala/fat_points/21-37/ We report the timings in the following table. We did also the defective case d = 21 as a sanity check. QUINTUPLE POINTS 25 Table 2. Timings in seconds for d = 21 . . . 37 from Theorem 17 d d 21 3539 29 53583 22 5137 30 87968 23 7557 31 107677 24 10911 32 143758 25 18020 33 194358 26 20535 34 255239 27 29089 35 378412 28 40221 36 511234 37 695840 Finally, we proved Theorem 18 in a similar manner, by using our programme fat points 13p.magma. Again, a slight modification of exact was needed in order to handle 13-degree points. The programme and the digital certificates can be found at http://www.science.unitn.it/~sala/fat_points/38-52/ The timings are reported in the following table Table 3. Timings in seconds for d = 38 . . . 52 from Theorem 18 d d 38 147495 46 323154 39 158191 47 373451 40 198248 48 460022 41 202834 49 517266 42 216555 50 717031 43 232417 51 783861 44 245465 52 1200723 45 325837 By observing the timings, we note an exponential behaviour (in d) for Table 1, approximately of behaviour 2d . This is easily explained, because the cost of the rank computation grows as d3 , but the number of cases to be examined grows exponentially. A similar behaviour can be seen in Table 2, where the times grow like (1.4)d . Indeed the reason why these latter computations are feasible lies in the significant cut in the number of cases to be observed. However, the real case thinning happens in Table 3, where the grows is only cubic in d. This fall from an exponential behaviour to a polynomial one can be explained only in a more-or-less constant number of cases to be considered (the cubic cost being unavoidable due to the cost of the rank computation). On the other hand, in Theorem 18 r can take only two values and the other integers are strictly bounded. As a further check, we computed the number of cases up to d = 100 and its maximum value is 405. Remark 30. We have used four Dell servers, each with two processors Intel Xeon X5460 at 3.16GHz (for a total of 32 processor cores) and with 32 GB’s of RAM (for a total of 128 GB). The underlying operating system has been Linux, kernel version 2.6.18-6-amd64. 6. The exceptions in degree 9 and 10 Our main theorem states that a general fat point scheme in P3 of multiplicity 5 has good postulation in degree d ≥ 11. Here we classify all the exceptional cases in degree 10 and 9. Let us consider first the case of degree 10. Let Y be a general union of w quintuple
points, x quartuple points, y triple points and z double points. Let N = 13 = 286. Then the linear system L = L3 (10; 5w , 4x , 3y , 2z ) has virtual 3 dimension vdim(L) = 286 − deg(Y ) − 1 where deg(Y ) = 35w + 20x + 10y + 4z and the expected dimension is max{vdim(L), −1}. Our programme checked all the cases with: • either w ≥ 1 and 286 − 3 ≤ deg(Y ) ≤ 286 + 34, • w = 7, 8 and deg(Y ) ≤ 286 + 34. 26 E. BALLICO, M. C. BRAMBILLA, F. CARUSO, AND M. SALA The programme found only nine cases of bad postulation, listed in the table below. In this table, we denote by e the expected dimension of the corresponding linear system, by r the rank of the matrix given by our construction, and by d the dimension of the linear system. Table 4. Exceptions in degree 10 w 9 8 8 8 8 8 7 7 7 x 0 1 0 0 0 0 2 1 2 y 0 0 1 1 0 0 0 2 0 z 0 0 1 0 2 1 1 0 0 min(deg(Y ), N ) 286 286 286 286 286 284 286 285 285 e -1 -1 -1 -1 -1 1 -1 0 0 r 285 284 285 283 284 282 284 284 280 d 0 1 0 2 1 3 1 1 5 From this computation we obtain the following classification: Theorem 31. In P3 a general union Y of w 5-points, x 4-points, y 3-points and z 2-points has good postulation in degree 10, except if the 4-tuple (w, x, y, z) is one of those listed in Table 6. Proof. If w = 0, then Y is a quartuple general fat point scheme and we already know by [4, 10] that it has good postulation in degree 10. We can thus assume w > 0. If Y is a general union of degree 283 ≤ deg(Y ) ≤ 320, our programme checked that there are no other cases of bad postulation, except for the ones listed in the table. Now if Y is a general union of degree deg(Y ) ≥ 321, then it contains a subscheme Y ′ of degree 286 ≤ deg(Y ′ ) ≤ 320 which has good postulation, except if Y is the union of w ≥ 10 quintuple points, where the only possible Y ′ is given by 9 quintuple points, which has bad postulation. On the other hand, by our computation we know that the dimension of the linear system L3 (10; 59 ) is 0. This means that as soon as we add a general simple point to Y ′ we immediately have an empty linear system. This implies that any union of w ≥ 10 quintuple points has good postulation. Now if Y has degree deg(Y ) ≤ 282, then it is contained in a scheme Y ′ of degree 283 ≤ deg(Y ′ ) ≤ 286 which has good postulation, obtained by adding only general double points. The only case we need to study more carefully are (w, x, y, z) = (8, 0, 0, 0), (7, 2, 0, 0), (8, 0, 1, 0), which correspond to subschemes of the exceptional cases with z > 0. We have checked directly that the first case (8 quintuple points) has good postulation, while the other two are exceptional cases. This completes the proof.  Some of the exceptional cases we found were already known, see e.g. [12] and [8]. Note that all the exceptions we found satisfy the conjecture of Laface-Ugaglia (see [12] and [13, Conjecture 6.3]). In the case of degree 9 we found many more exceptions, that we list in the following table. QUINTUPLE POINTS 27 Table 5. Exceptions in degree 9 w 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 3 3 3 x 0 0 0 0 0 0 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 3 6 5 5 y 1 1 0 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 0 0 0 2 0 1 1 z 1 0 3 2 1 0 1 0 1 0 3 2 1 0 6 5 4 3 2 1 0 0 0 1 0 min(deg(Y ), N ) 220 220 220 218 214 210 219 215 219 215 217 213 209 205 219 215 211 207 203 199 195 220 220 219 215 e -1 -1 -1 1 5 9 0 4 0 4 2 6 10 4 0 4 8 12 16 20 24 -1 -1 0 4 r 219 216 218 214 210 206 217 213 218 214 216 212 208 204 218 214 210 206 202 198 194 218 218 218 214 d 0 3 1 5 9 13 2 6 1 5 3 7 11 5 1 5 9 13 17 21 25 1 1 1 5 In this case we have tested all the configurations where w ≥ 1 and 220 − 3 ≤ deg(Y ) ≤ 220+34, and all the configurations with 1 ≤ w ≤ 6 and deg(Y ) ≤ 220−4. From our computational experiments we can deduce the following complete classification: Theorem 32. In P3 a general union Y of w 5-points, x 4-points, y 3-points and z 2-points has good postulation in degree 9, except if the 4-tuple (w, x, y, z) is one of those listed in Table 5. Proof. If w = 0, then Y is a quartuple general fat point scheme and it has good postulation in degree 10 by [4, 10]. We can thus assume w > 0. If Y is a general union of degree deg(Y ) ≤ 254, our programme checked that there are no other cases of bad postulation, except for the ones listed in the table. Now if Y is a general union of degree deg(Y ) ≥ 255, then it is easy to check that Y contains a subscheme Y ′ of degree 286 ≤ deg(Y ′ ) ≤ 320 which has good postulation.  Remark 33. Also in the case of degree 9 all the exceptions we found satisfy the conjecture of Laface-Ugaglia ([13, Conjecture 6.3]). The relevant computations can be found at http://www.science.unitn.it/~sala/fat_points/exceptions_9_10/ 28 E. BALLICO, M. C. BRAMBILLA, F. CARUSO, AND M. SALA 7. Further remarks In this final section we provide two remarks on the field characteristics and a direct consequence of Theorem 1. Since the result by Yang (Remark 13) is proved only for characteristic zero, we assume in this paper that char(K) = 0. However we underline that our proof of Theorem 1 can easily be adapted to any char(K) 6= 2, 3, 5. Hence the statement of Theorem 1 could immediately be generalized to any characteristic different from 2, 3, 5 as soon as we know that a general fat point scheme in P2 (F) of multiplicity 5 has good postulation in degree d ≥ 3m, for any field F with that characteristic, provided the result holds again for d = 11 in P3 (F). In positive characteristic the proof of Lemma 5 fails, since we cannot make use of Lemma 4. However, following the same outline as in the proof of Lemma 5 and recalling that a fat point always contains a simple point, it is not difficult to prove the following lemma. Lemma 34. Let F be an infinite field of any characteristic. Fix an integer d > 0. For any quadruple of non-negative integers (w, x, y, z), let Y (w, x, y, z) ⊂ P3 (F) denote a general union of w 5-points, x 4-points, y 3-points and z 2-points. If Y (w, x, y, z) has good postulation in degree d for any quadruple (w, x, y, z) such that     d+3 d+3 − 3 ≤ 35w + 20x + 10y + 4z ≤ +∆ 3 3 where  14    9 ∆= 5    2 if if if if w > 0 and x = y = z = 0, x > 0 and y = z = 0, y > 0 and z = 0, z>0 then any general quintuple fat point scheme has good postulation in degree d. A straightforward consequence of Theorem 1 is the following statement, whose proof is contained in Remark 3. Corollary 35. Fix non-negative integers w, x, y, z such that   14 . 35w + 20x + 10y + 4z ≥ 3 Let Y ⊂ P3 be a general union of w 5-points, x 4-points, y 3-points and z 2-points. Then Y has good postulation with respect to any degree. Acknowledgements We would like to thank an anonymous referee for his/her careful inspection of a previous version of this paper, where he spotted a nasty mistake. The first and second authors were partially supported by MIUR and GNSAGA of INdAM (Italy). The third and fourth author acknowledge support from the Provincia di Trento’s grant “Metodi algebrici per la teoria dei codici correttori e la crittografia”. The authors would like to thank M. Frego for his help in the computational part. QUINTUPLE POINTS 29 References [1] J. Alexander and A. Hirschowitz, La méthode d’Horace éclatée: application à l’interpolation endegrée quatre, Invent. Math. 107 (1992), 585–602. [2] J. Alexander and A. Hirschowitz, Polynomial interpolation in several variables, J. Alg. Geom. 4 (1995), no.2, 201–222. [3] J. Alexander and A. Hirschowitz, An asymptotic vanishing theorem for generic unions of multiple points, Invent. Math.140 (2000), no. 2, 303–325. [4] E. Ballico and M. C. 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Pure Appl. Algebra 151 (2000), no. 2, 173–195. [19] V. Strassen, Gaussian elimination is not optimal, Numer. Math. 13, 4 (1969), 354–356. [20] S. Winograd, On the multiplication of 2 × 2 matrices, Linear Algebra Appl. 4, 4 (1971), 381–388. [21] S. Yang, Linear systems in P2 with base points of bounded multiplicity, J. Algebraic Geom. 16 (2007), no. 1, 19–38. Dept. of Mathematics, University of Trento, 38123 Povo (TN), Italy E-mail address: [email protected] Dip. di Scienze Matematiche, Università Politecnica delle Marche, I-60131 Ancona, Italy E-mail address: [email protected] Dept. of Mathematics, University of Trento, 38123 Povo (TN), Italy E-mail address: [email protected] Dept. of Mathematics, University of Trento, 38123 Povo (TN), Italy E-mail address: [email protected]