scholarly journals Perverse Coherent Sheaves on Blow-ups at Codimension 2 Loci

2019 ◽  
Author(s):  
Naoki Koseki
Keyword(s):  
Moduli Space ◽  
Hilbert Scheme ◽  
Projective Variety ◽  
Blow Up ◽  
Smooth Projective Variety ◽  
Coherent Sheaves ◽  
Codimension Two ◽  
Higher Dimensional ◽  
Stable Sheaves ◽  
Closed Subvariety

Abstract Let $f \colon X \to Y$ be the blow-up of a smooth projective variety $Y$ along its codimension two smooth closed subvariety. In this paper, we show that the moduli space of stable sheaves on $X$ and $Y$ are connected by a sequence of flip-like diagrams. The result is a higher dimensional generalization of the result of Nakajima and Yoshioka, which is the case of $\dim Y=2$. As an application of our general result, we study the birational geometry of the Hilbert scheme of two points.

scholarly journals On the Singular Sheaves in the Fine Simpson Moduli Spaces of 1-dimensional Sheaves

2017 ◽  
Vol 60 (3) ◽  
pp. 522-535 ◽  
Author(s):  
Oleksandr Iena ◽  
Alain Leytem
Keyword(s):  
Projective Plane ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Blow Up ◽  
Line Bundles ◽  
Hilbert Polynomial ◽  
Stable Sheaves ◽  
Closed Subvariety

AbstractIn the Simpson moduli space M of semi-stable sheaves with Hilbert polynomial dm − 1 on a projective plane we study the closed subvariety M' of sheaves that are not locally free on their support. We show that for d ≥4 , it is a singular subvariety of codimension 2 in M. The blow up of M along M' is interpreted as a (partial) modification of M \ M' by line bundles (on support).

POSTNIKOV-STABILITY FOR COMPLEXES ON CURVES AND SURFACES

2012 ◽  
Vol 23 (02) ◽  
pp. 1250048 ◽  
Author(s):  
GEORG HEIN ◽  
DAVID PLOOG
Keyword(s):  
Moduli Space ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Derived Category ◽  
Stable Bundles ◽  
Coherent Sheaves ◽  
Curves And Surfaces

We present a novel notion of stable objects in the derived category of coherent sheaves on a smooth projective variety. As one application we compactify a moduli space of stable bundles using genuine complexes.

scholarly journals MODULI SPACE OF STABLE MAPS TO PROJECTIVE SPACE VIA GIT

2010 ◽  
Vol 21 (05) ◽  
pp. 639-664 ◽  
Author(s):  
YOUNG-HOON KIEM ◽  
HAN-BOM MOON
Keyword(s):  
Projective Space ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Projective Variety ◽  
Blow Up ◽  
Stable Maps ◽  
Homogeneous Polynomials ◽  
Birational Map

We compare the Kontsevich moduli space [Formula: see text] of stable maps to projective space with the quasi-map space ℙ( Sym d(ℂ2) ⊗ ℂn)//SL(2). Consider the birational map [Formula: see text] which assigns to an n tuple of degree d homogeneous polynomials f1, …, fn in two variables, the map f = (f1 : ⋯ : fn) : ℙ1 → ℙn-1. In this paper, for d = 3, we prove that [Formula: see text] is the composition of three blow-ups followed by two blow-downs. Furthermore, we identify the blow-up/down centers explicitly in terms of the moduli spaces [Formula: see text] with d = 1, 2. In particular, [Formula: see text] is the SL(2)-quotient of a smooth rational projective variety. The degree two case [Formula: see text], which is the blow-up of ℙ( Sym 2ℂ2 ⊗ ℂn)//SL(2) along ℙn-1, is worked out as a preliminary example.

RELATIVE GEOMETRIC INVARIANT THEORY AND UNIVERSAL MODULI SPACES

1996 ◽  
Vol 07 (02) ◽  
pp. 151-181 ◽  
Author(s):  
YI HU
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Invariant Theory ◽  
Hilbert Scheme ◽  
Geometric Invariant Theory ◽  
Geometric Invariant ◽  
Coherent Sheaves ◽  
Ample Cone

We expose in detail the principle that the relative geometric invariant theory of equivariant morphisms is related to the GIT for linearizations near the boundary of the G-effective ample cone. We then apply this principle to construct and reconstruct various universal moduli spaces. In particular, we constructed the universal moduli space over [Formula: see text] of Simpson’s p-semistable coherent sheaves and a canonical rational morphism from the universal Hilbert scheme over [Formula: see text] to a compactified universal Picard.

scholarly journals The Hodge cohomology and cubic equivalences

1984 ◽  
Vol 94 ◽  
pp. 1-41 ◽  
Author(s):  
Hiroshi Saito
Keyword(s):  
Complex Number ◽  
Algebraic Surface ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Chow Group ◽  
Geometric Genus ◽  
Singular Variety ◽  
Higher Dimensional ◽  
Hodge Cohomology ◽  
Complex Number Field

In 1969, Mumford [8] proved that, for a complete non-singular algebraic surface F over the complex number field C, the dimension of the Chow group of zero-cycles on F is infinite if the geometric genus of F is positive. To this end, he defined a regular 2-form ηf on a non-singular variety S for a regular 2-form η on F and for a morphism f: S → SnF, where SnF is the 72-th symmetric product of F, and he showed that ηf vanishes if all 0-cycles f(s), s ∈ S, are rationally equivalent. Roitman [9] later generalized this to a higher dimensional smooth projective variety V.

scholarly journals Cohomology bounds for sheaves of dimension one

2014 ◽  
Vol 25 (11) ◽  
pp. 1450103 ◽  
Author(s):  
Jinwon Choi ◽  
Kiryong Chung
Keyword(s):  
Projective Plane ◽  
Moduli Space ◽  
Hilbert Scheme ◽  
Closed Subset ◽  
Projective Variety ◽  
Global Section ◽  
Sharp Bounds ◽  
One Dimensional ◽  
Dimension One

We find sharp bounds on h0(F) for one-dimensional semistable sheaves F on a projective variety X. When X is the projective plane ℙ2, we study the stratification of the moduli space by the spectrum of sheaves. We show that the deepest stratum is isomorphic to a closed subset of a relative Hilbert scheme. This provides an example of a family of semistable sheaves having the biggest dimensional global section space.

scholarly journals THE MOTIVE OF THE HILBERT CUBE

2016 ◽  
Vol 4 ◽  
Author(s):  
MINGMIN SHEN ◽  
CHARLES VIAL
Keyword(s):  
Recent Work ◽  
Hilbert Scheme ◽  
Projective Variety ◽  
Abelian Varieties ◽  
Smooth Projective Variety ◽  
Hilbert Cube ◽  
Rational Map ◽  
Chow Rings ◽  
Hyperkähler Varieties

The Hilbert scheme $X^{[3]}$ of length-3 subschemes of a smooth projective variety $X$ is known to be smooth and projective. We investigate whether the property of having a multiplicative Chow–Künneth decomposition is stable under taking the Hilbert cube. This is achieved by considering an explicit resolution of the rational map $X^{3}{\dashrightarrow}X^{[3]}$. The case of the Hilbert square was taken care of in Shen and Vial [Mem. Amer. Math. Soc.240(1139) (2016), vii+163 pp]. The archetypical examples of varieties endowed with a multiplicative Chow–Künneth decomposition is given by abelian varieties. Recent work seems to suggest that hyperKähler varieties share the same property. Roughly, if a smooth projective variety $X$ has a multiplicative Chow–Künneth decomposition, then the Chow rings of its powers $X^{n}$ have a filtration, which is the expected Bloch–Beilinson filtration, that is split.

scholarly journals Height of exceptional collections and Hochschild cohomology of quasiphantom categories

2015 ◽  
Vol 2015 (708) ◽  
Author(s):  
Alexander Kuznetsov
Keyword(s):  
Projective Variety ◽  
Hochschild Cohomology ◽  
Smooth Projective Variety ◽  
Derived Category ◽  
Coherent Sheaves

AbstractWe define the normal Hochschild cohomology of an admissible subcategory of the derived category of coherent sheaves on a smooth projective variety

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