scholarly journals The integral cohomology of the Hilbert scheme of points on a surface

2020 ◽  
Vol 8 ◽  
Author(s):  
Burt Totaro
Keyword(s):  
Natural Number ◽  
Hilbert Scheme ◽  
Obstruction Theory ◽  
Projective Surface ◽  
Hilbert Schemes ◽  
Complex Projective Surface ◽  
Smooth Complex ◽  
Hilbert Scheme Of Points ◽  
Complex Projective ◽  
Torsion Free

Abstract We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme $X^{[n]}$ has torsion-free cohomology for every natural number n. This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas’s reduced obstruction theory for nested Hilbert schemes of surfaces.

scholarly journals The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces

2013 ◽  
Vol 149 (3) ◽  
pp. 481-494 ◽  
Author(s):  
François Charles ◽  
Eyal Markman
Keyword(s):  
Hilbert Scheme ◽  
K3 Surfaces ◽  
Cohomology Algebra ◽  
Hilbert Schemes ◽  
Projective Varieties ◽  
Hilbert Scheme Of Points ◽  
Complex Projective ◽  
Symplectic Varieties

AbstractWe prove the standard conjectures for complex projective varieties that are deformations of the Hilbert scheme of points on a K3 surface. The proof involves Verbitsky’s theory of hyperholomorphic sheaves and a study of the cohomology algebra of Hilbert schemes of K3 surfaces.

scholarly journals On the complex dynamics of birational surface maps defined over number fields

2016 ◽  
Vol 0 (0) ◽  
Author(s):  
Mattias Jonsson ◽  
Paul Reschke
Keyword(s):  
Number Field ◽  
Complex Dynamics ◽  
Energy Condition ◽  
Height Function ◽  
Number Fields ◽  
Projective Surface ◽  
Complex Projective Surface ◽  
Canonical Height ◽  
Dynamical Degree ◽  
Complex Projective

AbstractWe show that any birational selfmap of a complex projective surface that has dynamical degree greater than one and is defined over a number field automatically satisfies the Bedford–Diller energy condition after a suitable birational conjugacy. As a consequence, the complex dynamics of the map is well behaved. We also show that there is a well-defined canonical height function.

scholarly journals Stability of some vector bundles on Hilbert schemes of points on K3 surfaces

2021 ◽  
Author(s):  
Fabian Reede ◽  
Ziyu Zhang
Keyword(s):  
Moduli Space ◽  
Hilbert Scheme ◽  
Vector Bundles ◽  
K3 Surfaces ◽  
Hilbert Schemes ◽  
Connected Component ◽  
Stable Vector Bundles ◽  
Universal Family ◽  
Hilbert Scheme Of Points ◽  
Flat Family

AbstractLet X be a projective K3 surfaces. In two examples where there exists a fine moduli space M of stable vector bundles on X, isomorphic to a Hilbert scheme of points, we prove that the universal family $${\mathcal {E}}$$ E on $$X\times M$$ X × M can be understood as a complete flat family of stable vector bundles on M parametrized by X, which identifies X with a smooth connected component of some moduli space of stable sheaves on M.

scholarly journals Local finite-dimensional Gorenstein k-algebras having Hilbert function (1,5,5,1) are smoothable

2014 ◽  
Vol 13 (08) ◽  
pp. 1450056 ◽  
Author(s):  
Joachim Jelisiejew
Keyword(s):  
Hilbert Scheme ◽  
Hilbert Function ◽  
Hilbert Schemes ◽  
Gorenstein Algebras ◽  
Punctual Hilbert Scheme ◽  
Finite Dimensional ◽  
Hilbert Scheme Of Points

We consider the question of irreducibility of the Hilbert scheme of points ℋilbdℙn and its Gorenstein locus. This locus is known to be reducible for d ≥ 14. For d ≤ 11 the irreducibility of this locus was proved in the series of papers [G. Casnati and R. Notari, On the Gorenstein locus of some punctual Hilbert schemes, J. Pure Appl. Algebra 213(11) (2009) 2055–2074; On the irreducibility and the singularities of Gorenstein locus of the Punctual Hilbert scheme of degree 10, J. Pure Appl. Algebra 215(6) (2011) 1243–1254; Irreducibility of the Gorenstein locus of the Punctual Hilbert Scheme of degree 11, preprint (2012)] and Iarrobino conjectured that the irreducibility holds for d ≤ 13. In this paper, we prove that the subschemes corresponding to the Gorenstein algebras having Hilbert function (1, 5, 5, 1) are smoothable, i.e. lie in the closure of the locus of smooth subschemes. This result completes the proof of irreducibility of the Gorenstein locus of ℋilb12ℙn, see Theorem 2.

scholarly journals Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes

2015 ◽  
Vol 2015 (702) ◽  
pp. 1-40 ◽  
Author(s):  
Timo Schürg ◽  
Bertrand Toën ◽  
Gabriele Vezzosi
Keyword(s):  
Algebraic Geometry ◽  
Projective Variety ◽  
Obstruction Theory ◽  
Important Ingredient ◽  
Smooth Complex ◽  
Perfect Complexes ◽  
Derived Algebraic Geometry ◽  
Complex Projective Variety ◽  
Complex Projective

AbstractA quasi-smooth derived enhancement of a Deligne–Mumford stack 𝒳 naturally endows 𝒳 with a functorial perfect obstruction theory in the sense of Behrend–Fantechi. We apply this result to moduli of maps and perfect complexes on a smooth complex projective variety.ForWe give two further applications toAn important ingredient of our construction is a

scholarly journals Weyl and Zariski chambers on projective surfaces

2020 ◽  
Vol 32 (4) ◽  
pp. 1027-1037
Author(s):  
Krishna Hanumanthu ◽  
Nabanita Ray
Keyword(s):  
K3 Surfaces ◽  
Weyl Chamber ◽  
Projective Surface ◽  
Complex Projective Surface ◽  
Complex Projective ◽  
Projective Surfaces

AbstractLet X be a nonsingular complex projective surface. The Weyl and Zariski chambers give two interesting decompositions of the big cone of X. Following the ideas of [T. Bauer and M. Funke, Weyl and Zariski chambers on K3 surfaces, Forum Math. 24 2012, 3, 609–625] and [S. A. Rams and T. Szemberg, When are Zariski chambers numerically determined?, Forum Math. 28 2016, 6, 1159–1166], we study these two decompositions and determine when a Weyl chamber is contained in the interior of a Zariski chamber and vice versa. We also determine when a Weyl chamber can intersect non-trivially with a Zariski chamber.

scholarly journals HERMITIAN SURFACES OF CONSTANT HOLOMORPHIC SECTIONAL CURVATURE II

1992 ◽  
Vol 23 (2) ◽  
pp. 137-143
Author(s):  
TAKUJI SATO ◽  
KOUEI SEKIGAWA
Keyword(s):  
Sectional Curvature ◽  
Positive Constant ◽  
Holomorphic Sectional Curvature ◽  
Projective Surface ◽  
Complex Projective Surface ◽  
Complex Projective

The present paper ss a continuation of our previous work [7]. We shall prove that a compact Hernutian surface of pointwise positive constant holomorphic sectional curvature is biholomorphica.lly equivalent to a complex projective surface.

An example concerning Alexeev's boundedness results on log surfaces

1995 ◽  
Vol 118 (1) ◽  
pp. 65-69 ◽  
Author(s):  
Raimund Blache
Keyword(s):  
Intersection Theory ◽  
Projective Surface ◽  
Complex Projective Surface ◽  
Normal Complex ◽  
Quotient Singularities ◽  
Complex Projective

AbstractIn this note, we construct a sequence of l.t. surfaces (Xn)n ∈ ℕ such that KXn is ample for all n and such that (K2Xn)n ∈ ℕ is a strictly increasing series with limit equal to 1. This answers (in the affirmative) a question by Alexeev, cf. [Al], 11·1. Here, an l.t. surface is a normal complex projective surface with at most quotient singularities (which is the same as ‘at most log terminal singularities’). A main result of [Al] implies that it is impossible to find a sequence (Xn)n ∈ ℕ of l.t. surfaces with KXn ample for all n such that K2Xn is strictly decreasing. Although our construction is not too difficult, the example is new and has several interesting implications, see Section 4.Without further explanation, we use some fundamental tools concerning l.t. surfaces like Mumford's intersection theory or the notion of minimality; the reader should consult [Blb] and the references quoted there.

The arithmetic plurigenera of surfaces

1979 ◽  
Vol 85 (1) ◽  
pp. 25-31
Author(s):  
P. M. H. Wilson
Keyword(s):  
Smooth Surface ◽  
Projective Surface ◽  
Complex Projective Surface ◽  
Proper Morphism ◽  
The Family ◽  
Complex Projective ◽  
Positive Integers ◽  
Smooth Deformation

Let S0 be a complex projective surface with only isolated Gorenstein singularities (see Introduction to (12)). By Serre's criterion ((4), p. 185) this is equivalent to saying that S0 is normal and Gorenstein. By an algebraic smooth deformation of S0, we shall mean a flat, proper morphism of varieties, ρ: say, with fibre ρ−1(y0) = S0 for some y0 ∈ Y and with the general fibre ρ−1(y) = S being a smooth surface. In the paper (12), we studied such smooth deformations of S0 and in particular the behaviour of the plurigenera Pn of the surfaces in the family. The main result of (12) was the fact that Pn(S0) ≤ Pn(S) for all positive integers n, where the choice of the particular smooth surface was irrelevant by a result of Iitaka(5). To prove the above result we introduced what were called the arithmetic plurigenera of S0, which we define again below. In this paper we shall study more closely these arithmetic quantities, and in the process answer some of the questions posed in (11).

scholarly journals Incidence stratifications on Hilbert schemes of smooth surfaces, and an application to Poisson structures

2016 ◽  
Vol 27 (01) ◽  
pp. 1650006 ◽  
Author(s):  
Ziv Ran
Keyword(s):  
Poisson Structure ◽  
Hilbert Scheme ◽  
Smooth Surface ◽  
Smooth Curve ◽  
Smooth Surfaces ◽  
Poisson Structures ◽  
Hilbert Schemes ◽  
Hilbert Scheme Of Points

Given a smooth curve on a smooth surface, the Hilbert scheme of points on the surface is stratified according to the length of the intersection with the curve. The strata are highly singular. We show that this stratification admits a natural log-resolution, namely the stratified blowup. As a consequence, the induced Poisson structure on the Hilbert scheme of a Poisson surface has unobstructed deformations.

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