Noether-Lefschetz, space curves and mathematical instantons
type: Item
instance of: scholarly article
title: Noether-Lefschetz, space curves and mathematical instantons
zbMATH Open document ID: 0808.14009
DOI: 10.1007/BF01459741
published in: Mathematische Annalen
publication date: 1995-03-09T00:00:00Z
full work available at URL: https://eudml.org/doc/165177
review text: The author formulates, interrelates and in special cases proves conjectures pertaining to the three topics of the title. Let \(G_{(n,s)}\) denote the maximum genus of a smooth curve of degree \(n\) in \(\mathbb{P}^ 3\) (complex projective 3-space) not contained in a surface of degree less than \(s\), and let \(H_{n,G_{(n,s)}}\) denote the Hilbert scheme of smooth curves of degree \(n\) and genus \(G_{(n,s)}\) in \(\mathbb{P}^ 3\). Conjecture (H). If \({(s^ 2 + 4s + 6) \over 6} \leq n < {(s^ 2 + 4s + 6) \over 3}\) then \(H_{n,G_{(n,s)}}\) is pure dimensional of dimension \(4n\). This conjecture has been proven for certain special cases, e.g. \(s=2\). Next recall that an instanton bundle on \(\mathbb{P}^ 3\) is a rank 2 bundle \(F\) with \(c_ 1(F) = 0\), \(H^ 0(F) = H^ 1(F(-2)) = 0\). There is a well defined moduli space \(MI(0,c_ 2)\) of instantons with \(c_ 2(F) = c_ 2\). Conjecture \((\text{I}')\). Every component of \(MI (0,c_ 2)\) has dimension \(8c_ 2 -3.\) (Conjecture I is the conjecture that \(MI (0,c_ 2)\) is irreducible, and is known to imply \(\text{I}'\), where the notation. It has been proven for \(c_ 2 \leq 4.)\) A further technical conjecture, attributed to \textit{R. Hartshorne} [in: Vector bundles and differential equations, Proc., Nice 1979, Prog. Math. 7, 83-112 (1980; Zbl 0452.14005)] is: Conjecture (T). Let \(W\) be a component of \(M(0, c_ 2)\) and let \(F\) be a general element of \(W\). Then \(H^ 0 (F(t)) = 0\) for \(t \leq \sqrt {3c_ 2 + 1} - 2\). Let \(S(d)\) be the space of smooth surfaces of degree \(d\) in \(\mathbb{P}^ 3\) and let \(NL(d)\) be the Noether-Lefschetz locus, i.e., the subset of \(S(d)\) consisting of smooth surfaces whose Picard group is larger than \(\mathbb{Z}\). It is known that \(NL(d)\) is the union of countably many closed proper subsets of \(S(d)\), called components. A component of \(NL(d)\) is general if its codimension in \(S(d)\) is the expected value \(\left( \begin{smallmatrix} d - 1 \\ 3 \end{smallmatrix} \righ…
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 14J10
zbMATH DE Number: 562487
zbMATH Keywords: dimension of Hilbert scheme
zbMATH Keywords: dimension of moduli space of instantons
zbMATH Keywords: space curves
zbMATH Keywords: instanton bundle
zbMATH Keywords: Noether-Lefschetz locus
Wikidata QID: Q60143219
author: Angelo Felice Lopez
reviewed by: T. G. Berry
MaRDI profile type: Publication
cites work: Instantons and algebraic geometry
cites work: Irreducibility of the space of mathematical instanton bundles with rank 2 and \(c_2=4\)
cites work: Q4724762
cites work: Q3812361
cites work: The maximal rank conjecture for non special curves in \({\mathbb{P}}^ 3\)
cites work: Monads and moduli of vector bundles
cites work: General components of the Noether-Lefschetz locus and their density in the space of all surfaces
cites work: On the existence of components of the Noether-Lefschetz locus with given codimension
cites work: Halphen's gaps for space curves of submaximum genus
cites work: Sur le genre maximal des courbes gauches de degré d non sur une surface de degré s-1.
cites work: Hilbert scheme of smooth space curves
cites work: Q4027255
cites work: Koszul cohomology and the geometry of projective varieties. II
cites work: Components of maximal dimension in the Noether-Lefschetz locus
cites work: Genre des courbes de l'espace projectif
cites work: Q3331324
cites work: Stable vector bundles of rank 2 on \(P^ 3\)
cites work: Stable reflexive sheaves. II
cites work: Stable reflexive sheaves. III
cites work: Stable vector bundles and instantons
cites work: Nouvelles courbes de bon genre dans l'espace projectif. (New curves of good genus in projective space)
cites work: Cohomology of a general instanton bundle
cites work: A remark on the Hilbert scheme of smooth complex space curves
cites work: Irreducibility of a subscheme of the Hilbert scheme of complex space curves
cites work: Noether-Lefschetz theory and the Picard group of projective surfaces
cites work: Vector bundles on complex projective spaces
cites work: Composantes de petite codimension du lieu de Noether-Lefschetz. (Small codimension components of the Noether-Lefschetz locus)
cites work: Q3985695
cites work: Curves on surfaces with multiple line.
OpenAlex ID: W2010070741
description: scientific article; zbMATH DE number 562487