scholarly journals Gushel–Mukai varieties: Moduli

2020 ◽  
Vol 31 (02) ◽  
pp. 2050013 ◽  
Author(s):  
Olivier Debarre ◽  
Alexander Kuznetsov
Keyword(s):  
Moduli Space ◽  
Period Map ◽  
Moduli Stack ◽  
Git Quotient ◽  
Lagrangian Data ◽  
Comprehensive Study

We describe the moduli stack of Gushel–Mukai varieties as a global quotient stack and its coarse moduli space as the corresponding GIT quotient. The construction is based on a comprehensive study of the relation between this stack and the stack of so-called Lagrangian data defined in our previous works; roughly speaking, we show that the former is a generalized root stack of the latter. As an application, we define the period map for Gushel–Mukai varieties and construct some complete nonisotrivial families of smooth Gushel–Mukai varieties. In an appendix, we describe a generalization of the root stack construction used in our approach to the moduli stack.

scholarly journals THE GENERIC FIBRE OF MODULI SPACES OF BOUNDED LOCAL G-SHTUKAS

2021 ◽  
pp. 1-80
Author(s):  
Urs Hartl ◽  
Eva Viehmann
Keyword(s):  
Automorphism Group ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Function Field ◽  
Period Map ◽  
Generic Fibre ◽  
Divisible Groups

Abstract Moduli spaces of bounded local G-shtukas are a group-theoretic generalisation of the function field analogue of Rapoport and Zink’s moduli spaces of p-divisible groups. In this article we generalise some very prominent concepts in the theory of Rapoport-Zink spaces to our setting. More precisely, we define period spaces, as well as the period map from a moduli space of bounded local G-shtukas to the corresponding period space, and we determine the image of the period map. Furthermore, we define a tower of coverings of the generic fibre of the moduli space, which is equipped with a Hecke action and an action of a suitable automorphism group. Finally, we consider the $\ell $ -adic cohomology of these towers. Les espaces de modules de G-chtoucas locaux bornés sont une généralisation des espaces de modules de groupes p-divisibles de Rapoport-Zink, au cas d’un corps de fonctions local, pour des groupes plus généraux et des copoids pas nécessairement minuscules. Dans cet article nous définissons les espaces de périodes et l’application de périodes associés à un tel espace, et nous calculons son image. Nous étudions la tour au-dessus de la fibre générique de l’espace de modules, équipée d’une action de Hecke ainsi que d’une action d’un groupe d’automorphismes. Enfin, nous définissons la cohomologie $\ell $ -adique de ces tours.

scholarly journals Towards an enumerative geometry of the moduli space of twisted curves and rth roots

2008 ◽  
Vol 144 (6) ◽  
pp. 1461-1496 ◽  
Author(s):  
Alessandro Chiodo
Keyword(s):  
Moduli Space ◽  
Chern Character ◽  
Direct Image ◽  
Enumerative Geometry ◽  
Line Bundles ◽  
Stable Curves ◽  
Moduli Stack ◽  
Definition Of ◽  
Standard Techniques

AbstractThe enumerative geometry of rth roots of line bundles is crucial in the theory of r-spin curves and occurs in the calculation of Gromov–Witten invariants of orbifolds. It requires the definition of the suitable compact moduli stack and the generalization of the standard techniques from the theory of moduli of stable curves. In a previous paper, we constructed a compact moduli stack by describing the notion of stability in the context of twisted curves. In this paper, by working with stable twisted curves, we extend Mumford’s formula for the Chern character of the Hodge bundle to the direct image of the universal rth root in K-theory.

scholarly journals A Compactification of 3 via K3 Surfaces

2009 ◽  
Vol 196 ◽  
pp. 1-26 ◽  
Author(s):  
Michela Artebani
Keyword(s):  
Moduli Space ◽  
K3 Surfaces ◽  
Surjective Morphism ◽  
Period Map ◽  
Dimensional Ball

S. Kondō defined a birational period map between the moduli space of genus three curves and a moduli space of polarized K3 surfaces. In this paper we give a resolution of the period map, providing a surjective morphism from a suitable compactification of 3 to the Baily-Borel compactification of a six dimensional ball quotient.

scholarly journals The Brauer Group of the Universal Moduli Space of Vector Bundles Over Smooth Curves

2019 ◽  
Author(s):  
Roberto Fringuelli ◽  
Roberto Pirisi
Keyword(s):  
Moduli Space ◽  
Vector Bundles ◽  
Explicit Description ◽  
Complex Numbers ◽  
Brauer Group ◽  
Smooth Curves ◽  
Moduli Stack ◽  
Semistable Vector Bundles ◽  
Smooth Locus

Abstract We compute the Brauer group of the universal moduli stack of vector bundles on (possibly marked) smooth curves of genus at least three over the complex numbers. As consequence, we obtain an explicit description of the Brauer group of the smooth locus of the associated moduli space of semistable vector bundles, when the genus is at least four.

scholarly journals MODULI SPACES OF FRAMED SHEAVES ON CERTAIN RULED SURFACES OVER ELLIPTIC CURVES

2002 ◽  
Vol 13 (10) ◽  
pp. 1117-1151 ◽  
Author(s):  
THOMAS A. NEVINS
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Fixed Point Set ◽  
Explicit Formulas ◽  
Rational Homology ◽  
Moduli Stack ◽  
Framed Sheaves ◽  
Fixed Reference ◽  
Point Set ◽  
Projective Completion

Fix a ruled surface S obtained as the projective completion of a line bundle L on a complex elliptic curve C; we study the moduli problem of parametrizing certain pairs consisting of a sheaf ℰ on S and a map of ℰ to a fixed reference sheaf on S. We prove that the full moduli stack for this problem is representable by a scheme in some cases. Moreover, the moduli stack admits an action by the group C*, and we determine its fixed-point set, which leads to explicit formulas for the rational homology of the moduli space.

scholarly journals On the closure of the Hodge locus of positive period dimension

2021 ◽  
Author(s):  
B. Klingler ◽  
A. Otwinowska
Keyword(s):  
Moduli Space ◽  
Classical Result ◽  
Projective Variety ◽  
Abelian Varieties ◽  
Countable Union ◽  
Adjoint Group ◽  
Hodge Structures ◽  
Period Map ◽  
Tate Group ◽  
Irreducible Components

AbstractGiven $${{\mathbb {V}}}$$ V a polarizable variation of $${{\mathbb {Z}}}$$ Z -Hodge structures on a smooth connected complex quasi-projective variety S, the Hodge locus for $${{\mathbb {V}}}^\otimes $$ V ⊗ is the set of closed points s of S where the fiber $${{\mathbb {V}}}_s$$ V s has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for $${{\mathbb {V}}}^\otimes $$ V ⊗ is a countable union of closed irreducible algebraic subvarieties of S, called the special subvarieties of S for $${{\mathbb {V}}}$$ V . Under the assumption that the adjoint group of the generic Mumford–Tate group of $${{\mathbb {V}}}$$ V is simple we prove that the union of the special subvarieties for $${{\mathbb {V}}}$$ V whose image under the period map is not a point is either a closed algebraic subvariety of S or is Zariski-dense in S. This implies for instance the following typical intersection statement: given a Hodge-generic closed irreducible algebraic subvariety S of the moduli space $${{\mathcal {A}}}_g$$ A g of principally polarized Abelian varieties of dimension g, the union of the positive dimensional irreducible components of the intersection of S with the strict special subvarieties of $${{\mathcal {A}}}_g$$ A g is either a closed algebraic subvariety of S or is Zariski-dense in S.

scholarly journals Moduli of decorated swamps on a smooth projective curve

2015 ◽  
Vol 26 (10) ◽  
pp. 1550086 ◽  
Author(s):  
N. Beck
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Level Structure ◽  
Higgs Bundles ◽  
Smooth Projective Curve ◽  
Different Types ◽  
Git Quotient ◽  
Conic Bundles ◽  
Stability Concept

In order to unify the construction of the moduli space of vector bundles with different types of global decorations, such as Higgs bundles, framed vector bundles and conic bundles, A. H. W. Schmitt introduced the concept of a swamp. In this work, we consider vector bundles with both a global and a local decoration over a fixed point of the base. This generalizes the notion of parabolic vector bundles, vector bundles with a level structure and parabolic Higgs bundles. We introduce a notion of stability and construct the coarse moduli space for these objects as the GIT-quotient of a parameter space. In the case of parabolic vector bundles and vector bundles with a level structure our stability concept reproduces the known ones. Thus, our work unifies the construction of their moduli spaces.

Birational models of 𝓜2,2 arising as moduli of curves with nonspecial divisors

2021 ◽  
Vol 21 (1) ◽  
pp. 23-43
Author(s):  
Drew Johnson ◽  
Alexander Polishchuk
Keyword(s):  
Moduli Space ◽  
Moduli Of Curves ◽  
Moduli Space Of Curves ◽  
Stable Curves ◽  
Moduli Stack

Abstract We study birational projective models of 𝓜2,2 obtained from the moduli space of curves with nonspecial divisors. We describe geometrically which singular curves appear in these models and show that one of them is obtained by blowing down the Weierstrass divisor in the moduli stack of 𝓩-stable curves 𝓜 2,2(𝓩) defined by Smyth. As a corollary, we prove projectivity of the coarse moduli space M 2,2(𝓩).

scholarly journals p-adic period map for the moduli space of deformations of a formal group

2005 ◽  
Vol 288 (2) ◽  
pp. 445-462 ◽  
Author(s):  
Oleg Demchenko ◽  
Alexander Gurevich
Keyword(s):  
Moduli Space ◽  
Formal Group ◽  
Period Map
Sign in / Sign up

Export Citation Format

Share Document