Papers by geir ellingsrud
Inventiones Mathematicae, Jun 1, 1987
Although several authors have been interested in the Hilbert scheme Hilbd(lJ?2) parametrizing fin... more Although several authors have been interested in the Hilbert scheme Hilbd(lJ?2) parametrizing finite subschemes of length d in the projective plane { [Il ], [I2], (Fl ], [F2 ], [Br] among It seems natural to generali~e our: results to any toric E~ooth surface. Eowever, we give the :esul~s only for th~ . rational ruled sur£aces IP'n with a~ indication of tll~ nece~sary changes in the proofs. For sinplicity we \'>.'Or~<.: over ~he fieled of complex nqmbe:::;s, but with c.n appropriate interpretation of the word 11 homo1o9y'' our results remain v~lid over any base field. §1 Let IP 2 be the projective plane over c. Jrc:>r a~y po&:JitiV+l integer d, let Hilbd(P 2 } qenote the ailbert echeme par~ne trizing finite subschemes of f22 o:e 1e:n~h Q.. It 4\? dep,otes the complement of a, line in IP 2 , .let Jii.~.:}:ld(D>.. 2 ) denote the open subscheme of Hilbd(~2) ~orresponding to sUbschemas with $Upport in ~ 2 • Fu;r;'thermore let iiilbd(h\2, 0) be the closeq ~up.scnerne of Hilbd(~. 2 } pax-ametrizing subschemas suppox-t~~ in the <:>rigin. For any complex variety X, let H* (X) b~ the :ao;rel..,.t""..oore homology of X (homology with locally finite supports). By the i-th Betti number bi(X) we shall mean the ~an~ o~ tbe finitely generated abelian group Hi (X). Let ;dX) = r (-1) ibi (X) be the Euler-Poincare characteristic of x. As usual, A*(X) is the Chow group of X, and cl:A*(X)"-+ f.t*(:X:) is the cyclE! map (see (Fu] ch. 19.1). If m .and n are non-negativ' integers, let P(:m,n) denote the number of sequences n~b 0 >b 1 ) ••• ~bm = 0 sucn that Lbi = m. If n:>m, then P (m, p) = P (m), the numbel:' of pal;"titions of m. Let P(m,n) = 0 if o or n is negative* ' )' \' * ~s an isomorphism.
Transactions of the American Mathematical Society, 1998
We compute the intersection number between two cycles A and B of complementary dimensions in the ... more We compute the intersection number between two cycles A and B of complementary dimensions in the Hilbert scheme H parameterizing subschemes of given finite length n of a smooth projective surface S. The (n+ 1)cycle A corresponds to the set of finite closed subschemes the support of which has cardinality 1. The (n -1)-cycle B consists of the closed subschemes the support of which is one given point of the surface. Since B is contained in A, indirect methods are needed. The intersection number is A.B = (-1) n-1 n, answering a question by H. Nakajima.
Quarterly Journal of Mathematics, Sep 1, 1998
Mathematische Zeitschrift, Jun 1, 1983
Mathematische Annalen, Mar 1, 1981
... Geir Ellingsrud and Stein Arild Stromme* Institute of Mathematics, University of Oslo, Blinde... more ... Geir Ellingsrud and Stein Arild Stromme* Institute of Mathematics, University of Oslo, Blindern, Oslo 3, Norway ... In particular, if E is a rank-2 bundle on X with cl(E ) = 0, and L =X is a line, then EL _-__ (gL(~)O6OL(- ?) for some integer 7 = 7(L)&amp;gt;0. Following Barth [1] we say that L is ...
arXiv (Cornell University), Sep 29, 1994
Counting twisted cubic curves on general complete intersections. By G. Ellingsrud and S. A. Strøm... more Counting twisted cubic curves on general complete intersections. By G. Ellingsrud and S. A. Strømme. 14 pages, amslatex 1.1 This paper has been rendered obsolete by our newer eprint 9411005 "Bott's formula and enumerative geometry", which is a considerably expanded version of the same paper, in spite of the change of titles. Please download 9411005 instead, and update possible references.
CRC Press eBooks, May 10, 2023
ABSTRACT We compute the Donaldson numbers $q_{17}(CP^2)=2540$ and $q_{21}(CP^2)=233208$.
Cambridge University Press eBooks, Jul 30, 1992
... F.-LETIZIA M. On the Betti numbers of the moduli space of stable bundles of rank two on a cur... more ... F.-LETIZIA M. On the Betti numbers of the moduli space of stable bundles of rank two on a curve 92 CILIBERTO C.-MIRANDA ... Trieste was made possible by the supports of the&amp;amp;quot; Consiglio Nazionale délie Ricerche&amp;amp;quot;(CNR), the regional authorities of&amp;amp;quot; Friuli-Venezia Giulia&amp;amp;quot; and the ...
Springer eBooks, 1987
ABSTRACT Without Abstract
Crelle's Journal, Aug 1, 1993
Inventiones Mathematicae, Feb 1, 1989
§ 0. Introduction. Let k be any algebraically closed field, and denote by M = M(-1,n) the fine mo... more § 0. Introduction. Let k be any algebraically closed field, and denote by M = M(-1,n) the fine moduli space of stable vector 2 2 bundles on JP = JPk of rank 2 with Chern classes c 1 = -1 and c 2 = n. [3, thm 7.17]. If n~O, then M = 0, and if n = 1, M = Speck. In this paper we prove the following Theorem Suppose n.:::,2. Then PicM is generated by two elements and c with one relation nc = 2m. In particular, Pic M = Z if n is odd, and PicM = Z $ Z/2Z if n is even. Remark: m and c a.re defined in § 2. Remark: Le Potier [2] has computed Pic M(O,n) in the case k =G.:, using the technique of monads. The proof goes along the following lines: First we find a decomposition of M into the union of three locally closed subsets, M 0 , M 1 , and ~2 such that M 0 is open and dense in M, the closure of M 1 has codimension 1, and ~2 is closed of codimension 2. We give complete descriptions of M 0 and M 1 , in particular, we compute their Picard groups. It turns out that this, together with the restriction map Pic M -.> Pic M 1 , is sufficient to determine Pic I'1 completely.
arXiv (Cornell University), Apr 15, 1997
Let X be a smooth projective surface, E a locally free sheaf of rank r ≥ 1 on X, and let ℓ ≥ 1 be... more Let X be a smooth projective surface, E a locally free sheaf of rank r ≥ 1 on X, and let ℓ ≥ 1 be an integer. Quot(E, ℓ) denotes Grothendieck's quotient scheme that parametrises all surjections E → T , where T is a zero-dimensional sheaf of length ℓ, modulo automorphisms of T . Sending a quotient E → T to the point x∈X ℓ(T x )x in the symmetric product S ℓ (X) defines a morphism π : Quot(E, ℓ) → S ℓ (X) . It is the purpose of this note to prove the following theorem: Theorem 1 -Quot(E, ℓ) is an irreducible scheme of dimension ℓ(r + 1). The fibre of the morphism π : Quot(E, ℓ) → S ℓ (X) over a point x ℓ x x is irreducible of dimension x (rℓ x -1). If r = 1, i.e. if E is a line bundle, then Quot(E, ℓ) is isomorphic to the Hilbert scheme Hilb ℓ (X). For this case, the first assertion of the theorem is due to Fogarty [5], whereas the second assertion was proved by Briançon . For general r ≥ 2, the first assertion of the theorem is a result due to J. Li and D. Gieseker [8], . We give a different proof with a more geometric flavour, generalising a technique from Ellingsrud and Strømme . The second assertion is a new result for r ≥ 2.
arXiv (Cornell University), Oct 7, 1994
Annales Scientifiques De L Ecole Normale Superieure, 1975
Sur le schéma de Hilbert des variétés de codimension 2 dans P e à cône de Cohen-Macaulay Annales ... more Sur le schéma de Hilbert des variétés de codimension 2 dans P e à cône de Cohen-Macaulay Annales scientifiques de l'É.N.S. 4 e série, tome 8, n o 4 (1975), p. 423-431 <http © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1975, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » () implique l'accord avec les conditions générales d'utilisation (). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
§ 0. Introduction Let X = JP~ denote the projective 3-space over an algebraically closed field k ... more § 0. Introduction Let X = JP~ denote the projective 3-space over an algebraically closed field k of characteristic zero. Given an integer n, denote by M(n) the moduli space for stable rank-2 vector bundles on X with Chern classes c~ = 0 and c 2 = n, see . In his survey article [~0], M. Schneider asks the following question: Are M(3) and M(4) nonsingular, and do they have only two components? In this paper we answer this question affirmatively for M(3), and we also prove that both components are rational~ Our main tool in the proof will be a careful study of the restriction of a bundle to all lines through a fixed point P in X. By a theorem of Grothendieck [4] any vector bundle on a projective line is a direct sum of lineb~~dles. In particular, if E is a rank-2 bundle on X with c~(E) = o, and LeX is a line, then ~~ <9L(y) $$L(-y) for some integer y = y (L) ~ 0. Following Barth [ 1 ] we say that L is a jumping ~ for E if y(L) I 0. A jumping line L is said to be multiple if y(L) > '1. The well-known theorem of Gra.~rt Mulich [~] states that if E is stable (in this case this is equivalent to H 0 (X,E) = 0), then the general line is not a jumping line. If P E X(k) is a closed point, denote by Mp (n) the open subscheme of M(n) parametrizing bundles E satisfying the following two conditions: There exists a non-jumping line for E through P. (ii) There are no multiple jumping lines for E through P.
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Papers by geir ellingsrud