Papers by Edoardo Ballico
Abstract: In this paper we construct, for every $ n $, smooth varieties of general type of dimens... more Abstract: In this paper we construct, for every $ n $, smooth varieties of general type of dimension $ n $ with the first $\ lfloor\ frac {n-2}{3}\ rfloor $ plurigenera equal to zero. Hacon-McKernan, Takayama and Tsuji have recently shown that there are numbers $ r_n $ such that $\ forall r\ ge r_n $, the $ r-$ canonical map of every variety of general type of dimension $ n $ is birational. Our examples show that $ r_n $ grows at least quadratically as a function of $ n $.
We find some ranges for the $4$-tuples of integers $(d, g, n, r)$ for which there is a smooth con... more We find some ranges for the $4$-tuples of integers $(d, g, n, r)$ for which there is a smooth connected non-degenerate curve of degree $d$ and genus $g$, which is $k$-normal for every $k\leq r$.
Fix integers d n 3. Here we show the existence of a nodal and connected tree-like (i.e. with line... more Fix integers d n 3. Here we show the existence of a nodal and connected tree-like (i.e. with lines as irreducible components) curve Y Pn such that pa(Y ) = 0, deg(Y ) = d and Y with maximal rank. If either n 6= 3 or d is not " exceptional " we prove that we may take Y of
Here we study the postulation of su-ciently general reducible con- nected nodal curves T ‰ Pn, n ... more Here we study the postulation of su-ciently general reducible con- nected nodal curves T ‰ Pn, n ‚ 3, such that every irreducible component of T is a line. We will also consider the postulation of the general hyperplane section of T and study reducible connected curves Y which are union of a rational normal curve of Pn and deg(Y
We extend to higher Gaussian maps two classical theorems by Giuseppe Gherardelli and Beniamino Se... more We extend to higher Gaussian maps two classical theorems by Giuseppe Gherardelli and Beniamino Segre which give a sharp lower bound on the rank and characterize the borderline case.
Topology and its Applications, 2012
ABSTRACT Consider a simply connected, smooth, projective, complex surface X. Let be the moduli sp... more ABSTRACT Consider a simply connected, smooth, projective, complex surface X. Let be the moduli space of framed irreducible anti-self-dual connections on a principal SU(2)-bundle over X with second Chern class k>0, and let be the corresponding space of all framed connections, modulo gauge equivalence. A famous conjecture by M. Atiyah and J. Jones says that the inclusion map induces isomorphisms in homology and homotopy through a range that grows with k.In this paper, we focus on the fundamental group, π1. When this group is finite or polycyclic-by-finite, we prove that if the π1-part of the conjecture holds for a surface X, then it also holds for the surface obtained by blowing up X at n points. As a corollary, we get that the π1-part of the conjecture is true for any surface obtained by blowing up n times the complex projective plane at arbitrary points. Moreover, for such a surface, the fundamental group is either trivial or isomorphic to Z2.
Linear Algebra and its Applications, 2013
For any irreducible non-degenerate variety X ⊂ P r , we relate the dimension of the s-th secant v... more For any irreducible non-degenerate variety X ⊂ P r , we relate the dimension of the s-th secant varieties of the Segre embedding of P k × X to the dimension of the (k, s)-Grassmann secant variety GS X (k, s) of X. We also give a criterion for the s-identifiability of X.
Rendiconti del Circolo Matematico di Palermo, 2003
EDOARDO BALLICO - CLAUDIO FONTANARI Let V (g, x, k, y) be the set of all pairs (X, F), where X is... more EDOARDO BALLICO - CLAUDIO FONTANARI Let V (g, x, k, y) be the set of all pairs (X, F), where X is an integral projective nodal curve with p a (X ) = g and card(Sing( X )) = x and F is a rank 1 torsion free sheaf on X with deg(F) = k , card(Sing(F)) = y and h 0 (X, F) ≥ 2. Here we study a general (X, F) ∈ V (g, x, k, y) and in particular the Brill-Noether theory of X and the scrollar invariants of F .
Let X be a smooth genus g curve equipped with a simple morphism f: X -> C, where C is either t... more Let X be a smooth genus g curve equipped with a simple morphism f: X -> C, where C is either the projective line or more generally any smooth curve whose gonality is computed by finitely many pencils. Here we apply a method developed by Aprodu to prove that if g is big enough then X satisfies both Green and Green-Lazarsfeld conjectures. We also partially address the case in which the gonality of C is computed by infinitely many pencils.
Annali Dell'universita' Di Ferrara, 2001
Sunto In questo lavoro si dimostra il seguente teorema. Teorem 1.1.Sia X una curva proiettiva ri... more Sunto In questo lavoro si dimostra il seguente teorema. Teorem 1.1.Sia X una curva proiettiva ridotta e irriducibile di genere aritmetico g e k≥4 un intero. Si supponga l'esistenza di L ε Pick (X) con h 0 (X, L)=2 e L generato. Si fissi un fascio senza torsione di rango uno M su X con h0 (X, M)=r++1≥2, h1 (X, M) ≧2 e M generato dalle sue sezioni globali. Si ponga d≔deg(M) e s≔max{n≧0:h 0(X, M ⊗(L*)⊗n)>0}. Allora si verifica uno dei casi seguenti: (a) M≊L ⊗r; (b) M è il sottofascio di ω X⊗(L*)⊗t, t:=g−d+r−1 generato da H0 (X, ωX⊗(L*)⊗t); (c) esiste un fascio senza torsione di rango un F su X con 1≦h 0 (X, F) ⊗8 ⊗ F. Inoltre, se si fissa un intero m con 2≦m≦k−2 e si suppone r#(s+1) k−(ns+n+1) per ogni 2≦n≦m, si ottiene h 0 (X, F)≦k−m−1. Si ricavano anche altre maggiorazioni suh 0,(X, F).
Annali Dell'universita' Di Ferrara, 2004
Sunto Perd≥3g et 1≤s≤[g/2], si studiano gli stratiN d,g(s) delle curveC di ℙ3 aventi gradod e ge... more Sunto Perd≥3g et 1≤s≤[g/2], si studiano gli stratiN d,g(s) delle curveC di ℙ3 aventi gradod e genereg il cui fibrato normaleN C è stabile con grado di stabilità (intero di Lange-Narasimhan) σ(N C)=2s. Si prova cheN d, g(s) ha una componente irriducibile della giusta dimensione la cui curva generale ha un fibrato normale avente il numero di sottofibrati massimali che ci si aspetta. Consideriamo anche il caso semistabile (s=0), ottenendo risultati simili. Vengono usate deformazioni di curve e di fibrati, si studiano i fibrati normali di curve riducibili.
Monatshefte für Mathematik, 2002
Let S be a smooth projective surface. Here we study the conditions imposed to curves of a ®xed ve... more Let S be a smooth projective surface. Here we study the conditions imposed to curves of a ®xed very ample linear system by a general union of types of singularities when most of connected components of are ordinary double points. This problem is related to the existence of``good'' families of curves on S with prescribed singularities, most of them being nodes, and to the regularity of their Hilbert scheme.
Mathematische Zeitschrift, 2012
Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically cl... more Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the s-th secant variety of the d-uple Veronese embedding of P m into P ( m+d d )−1 but that its minimal decomposition as a sum of d-th powers of linear
Journal of Symbolic Computation, 2004
By using a computer we are able to pose a conjecture for the expected number of generators of the... more By using a computer we are able to pose a conjecture for the expected number of generators of the ideal of a non-special general irreducible curve in P r with degree d, genus g, for d ≥ r + g. We prove the conjecture for C of degree d ≤ 60.
Journal of Pure and Applied Algebra, 2009
ABSTRACT
Journal of Pure and Applied Algebra, 2008
Let Z be a fat point scheme in P 2 supported on general points. Here we prove that if the multipl... more Let Z be a fat point scheme in P 2 supported on general points. Here we prove that if the multiplicities are at most 3 and the length of Z is sufficiently high then the number of generators of the homogeneous ideal I Z in each degree is as small as numerically possible. Since it is known that Z has maximal Hilbert function, this implies that Z has the expected minimal free resolution.
Journal of Pure and Applied Algebra, 2006
We prove various properties of varieties of special linear systems on double coverings of hyperel... more We prove various properties of varieties of special linear systems on double coverings of hyperelliptic curves. We show and determine the irreducibility, generically reducedness and singular loci of the variety W r d for bi-elliptic curves and double coverings of genus two curves. Similar results for double coverings of hyperelliptic curves of genus h ≥ 3 are also presented.
Journal of Pure and Applied Algebra, 2009
We study the postulation of a general union Y of double, triple, quartuple and quintuple points o... more We study the postulation of a general union Y of double, triple, quartuple and quintuple points of P 3 . 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.
Journal of Pure and Applied Algebra, 2010
Let L be a very ample line bundle of degree d on a general curve X of genus g ≥ 2. Here we prove ... more Let L be a very ample line bundle of degree d on a general curve X of genus g ≥ 2. Here we prove that if d > g + max √ 6g, g h 1 (L)+2 + 1 then L is globally generated, i.e. L embeds X as a projectively normal curve in PH 0 (L).
Journal of Pure and Applied Algebra, 2009
Here we define the concept of Qregularity for coherent sheaves on quadrics. In this setting we pr... more Here we define the concept of Qregularity for coherent sheaves on quadrics. In this setting we prove analogs of some classical properties. We compare the Qregularity of coherent sheaves on Q n ⊂ P n+1 with the Castelnuovo-Mumford regularity of their extension by zero in P n+1 . We also classify the coherent sheaves with Qregularity −∞. We use our notion of Qregularity in order to prove an extension of Evans-Griffiths criterion to vector bundles on Quadrics. In particular we get a new and simple proof of the Knörrer's characterization of ACM bundles.
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Papers by Edoardo Ballico