Abstracts

Hülia Argüz (University of Georgia)

Moduli spaces in algebraic geometry I: curves

This is an introductory talk about moduli spaces in algebraic geometry, with an emphasis on the moduli space of stable curves. In particular, we will discuss compactifications of the moduli spaces of curves and review birational geometric aspects.

Harold Blum (University of Utah)

Moduli of Fano varieties with complements

While the theories of KSBA-stability and K-stability have been successful in constructing compact moduli spaces of canonically polarized varieties and Fano varieties, respectively, the case of K-trivial varieties remains less well understood. I will discuss a new approach to this problem in the case of K-trivial pairs (X,D), where X is a Fano variety and D is an anticanonical Q-divisor, in which we consider all slc degenerations. In the case when X is a degeneration of P2, this approach gives rise to an interesting family of moduli spaces. This is joint work with K. Ascher, D. Bejleri, K. DeVleming, G. Inchiostro, Y. Liu, X. Wang.

Pierrick Bousseau (University of Georgia)

Moduli spaces in algebraic geometry II: surfaces

A natural generalization of a stable curve to dimension two is given by a KSBA stable surface. In this expository talk we will discuss the moduli space of KSBA stable surfaces. As an explicit example, we will describe the KSBA moduli space for toric surfaces.

Kristin DeVleming (University of Massachusetts, Amherst)

A question of Mori and families of plane curves

Abstract: Consider a smooth family of hypersurfaces of degree d in Pn+1.  An old question of Mori is: when is every smooth limit of this family also a hypersurface? While it is easy to construct examples where the answer is "no" when the degree d is composite, there are no known examples when d is prime and n>2.  While we will pose this as a conjecture, there are counterexamples when n=1 or 2. In this talk, we will propose a re-formulation of the conjecture that explains the failure in low dimensions, provide results in dimension one, and discuss a general approach to the problem using moduli spaces of pairs. This is joint work with David Stapleton.

Yunfeng Jiang (University of Kansas)

The virtual fundamental class for the moduli space of general type surfaces

Abstract:  Sir Simon Donaldson conjectured that there should exist a virtual fundamental class on the moduli space of surfaces of general type inspired by the geometry of complex structures on the general type surfaces. In this talk I will present a method to construct the virtual fundamental class on the moduli stack of  lci (locally complete intersection) covers over the moduli stack of general type surfaces with only semi-log-canonical singularities. A tautological invariant is defined by taking the integration of the power of the first Chern class of the CM line bundle over the virtual fundamental class. This can be taken as a generalization of the tautological invariants on the moduli space of stable  curves to the moduli space of stable surfaces.  If time permits, we also talk about the  possible methods to construct a virtual fundamental class on the Alexeev moduli space of stable maps from semi-log-canonical surfaces to projective varieties.

Radu Laza (Stony Brook)

Deformations of mildly singular Calabi-Yau varieties

The well-known Bogomolov-Tian-Todorov theorem says that the deformations of Calabi-Yau manifolds are unobstructed. The unobstructedness of deformations continues to hold Calabi-Yau varieties with ordinary nodal singularities (Kawamata, Ran, Tian), but surprisingly the smoothability of such varieties is subject to topological constrains. These obstructions to the existence of smoothings are linear in dimension 3 (Friedman), and non-linear in higher dimensions (Rollenske-Thomas).

In this talk, I will give vast generalizations to both the unobstructedness of deformations for mildly singular Calabi-Yau varieties, and to the constraints on the existence of smoothings for certain classes of singular Calabi-Yau varieties. Additionally, I will establish the proper context for these results: the Hodge theory of degenerations with prescribed singularities (specifically higher rational and higher Du Bois singularities).

This is joint work with Robert Friedman.

Rita Pardini (University of Pisa)

Stable I-surfaces of index 2 and generalized spin curves of genus 2

An I-surface is a complex projective surface with K2=1​​, h2(O)=2​​ and ample canonical class. Gorenstein stable I-surfaces are hypersurfaces of degree 10 in P(1,1,2,5). In order to study 2-Gorenstein  I-surfaces we introduce generalized Gorenstein spin curves, namely  pairs (C,L)​​ where C​​ is a Gorenstein curve with ample canonical class and L​​ is a torsion free rank 1 sheaf on C​​ with χ(L)=0​​ admitting a generically injective map L⊗ L→ωC​​. We obtain a complete classification  of such pairs with C reduced of genus 2 and derive from it  the  classification of stable I-surfaces of index 2 with a reduced canonical curve.

This is joint work in progress with S.Coughlan, M.Franciosi and S.Rollenske.

Junliang Shen (Yale)

Moduli of sheaves and BPS invariants

BPS numbers are curve-counting invariants associated with Calabi-Yau 3-folds. In the last a few decades, a lot of progress has been made to understand structures of these invariants better. In this talk, I will first explain how/why many classical moduli spaces, including certain moduli of sheaves on surfaces, and moduli of Higgs bundles on curves show up naturally in the BPS story. Then I will discuss some recent progress on exploring geometric structures of these classical moduli spaces that underly the BPS invariants.

Minitalks

Deng, Haohua (Duke University) Theories on extending period maps

Genlik, Deniz (Ohio State University) Crepant Resolution Conjecture for Cn/Zn

Heath, Bailey (University of South Carolina) Representation Dimensions of Algebraic Tori

Lank, Pat (University of South Carolina) High Frobenius pushforwards generate the bounded derived category

Lee, Jae Hwang (Colorado State University) A quantum H*(G)-module via quasi-map invariants

Mushunje, Leonard (Columbia University) Topological manifolds for statistical learning

Smith, Jonathan (University of South Carolina) Automorphisms of del Pezzo Surfaces over Number Fields

Tayou, Salim (Harvard University) Mixed mock modularity of special divisors

Tighe, Benjamin (University of Illinois at Chicago) The LLV Algebra for Singular Symplectic Varieties

Venkatesh, Sridhar (University of Michigan) Higher derived pushforwards of log differentials on toric varieties

Shen, Wanchun (Harvard University) Higher Du Bois and higher rational singularities for cones

Xiang, Fei (University of California, Irvine) On the Crepant Resolutions and Quiver Moduli Spaces

Zhao, Junyan (University of Illinois at Chicago) Moduli space of genus six curves and K-stability