September 7: Takumi Murayama (Purdue University)
Location: REC 302
Title: Pure subrings of Du Bois singularities are Du Bois singularities
Abstract: Let R → S be a pure map of rings. If S is regular, then it is known that R has rational singularities, and hence is Cohen-Macaulay. This result applies in particular to inclusions of rings of invariants by linearly reductive groups, split maps, and faithfully flat maps.
In this talk, I will discuss my recent work (joint with Charles Godfrey) showing that if R and S are essentially of finite type over the complex numbers, and S has Du Bois singularities, then R has Du Bois singularities. Our result is new even when R → S is faithfully flat. As a consequence, under the same hypotheses on R → S, if S has log canonical type singularities and the canonical divisor on R is Cartier, then R has log canonical singularities.
September 14: Shiji Lyu (Princeton University)
Location: REC 302
Title: Some properties of splinters and birational derived splinters
Abstract: We will discuss the notions of splinters and birational derived splinters. They are related to various interesting notions of singularities, but less are known about them. We will discuss some basic properties of those notions including behavior under limit and etale extensions. Then we will discuss some more advanced properties, one of which involves ultrapower to prove.
September 21: Raman Parimala (Emory University)
Location: REC 302 (Special time: 2:30-3:30PM)
Title: Pencils of quadrics and hyperelliptic curves
Abstract: Connections between the complex geometry of a hyperelliptic curve C and the internal geometry of the base locus of the associated pencil of quadrics are classical and trace back to Andre Weil. There is a rational description of the moduli space of rank 2 stable bundles with odd determinant on a smooth hyperelliptic curve C of genus g in terms of the Grassmannian of g-1 dimensional linear subspaces contained in the base locus of the associated pencil of quadrics due to Ramanan. We explain a twist of this construction which leads to connections between period index bounds for the unramified Brauer classes on K(C), K being a totally imaginary number field and the existence of rational points on the Grasmannians in the associated pencil of quadrics (joint work with Jaya Iyer).
September 28: Yilong Zhang (Purdue University)
Location: REC 302 and Zoom (https://purdue-edu.zoom.us/j/97786675764?pwd=ZUxkQzFsck5IaFBBZnhiOVNTZys4UT09)
Title: Cubic threefold and vanishing cycles on its hyperplane sections
Abstract: For a complex projective manifold X, a vanishing cycle is a topological sphere on a smooth hyperplane section that is contracted to a point as the hyperplane section deforms and becomes tangent to X. What is a vanishing cycle on a singular hyperplane section? We will try to answer the question in the case when X is a smooth cubic hypersurface of the complex projective four-space. In particular, we compactify the parameter space of vanishing cycles on smooth hyperplane sections and interpret the boundary points by the Hilbert scheme of X and singularities on cubic surfaces.
October 5: Jennifer Li (Princeton University)
Location: Zoom (https://purdue-edu.zoom.us/j/97786675764?pwd=ZUxkQzFsck5IaFBBZnhiOVNTZys4UT09)
Title: A cone conjecture for log Calabi-Yau surfaces
Abstract: In 1993, Morrison conjectured that the automorphism group of a Calabi-Yau 3-fold acts on its nef cone with a rational polyhedral fundamental domain. In this talk, I will discuss a version of this conjecture for log Calabi-Yau surfaces. In particular, for a generic log Calabi-Yau surface with singular boundary, the monodromy group acts on the nef effective cone with a rational polyhedral fundamental domain. In addition, the automorphism group of the unique surface with a split mixed Hodge structure in each deformation type acts on the nef effective cone with a rational polyhedral fundamental domain.
October 19: Mihai Fulger (University of Connecticut)
Location: REC 302
Title: Positivity vs. semi-stability for bundles with vanishing discriminant
Abstract: On curves a bundle is semistable iff its normalization satisfies the nef positivity condition. Furthermore, for semistable bundles nefness (resp. ampleness) is equivalent to the nefness (resp. ampleness) of its determinant. These are classical results going back to work of Hartshorne. In higher dimension generalizations hold under an additional necessary condition, the vanishing of the discriminant. We present algebraic proofs of this. This is in joint work with Adrian Langer.
October 26: Joseph Knight (Purdue University)
Location: REC 302 and Zoom (https://purdue-edu.zoom.us/j/97786675764?pwd=ZUxkQzFsck5IaFBBZnhiOVNTZys4UT09)
Title: The EKL-degree of a Weyl cover.
Abstract: For a field K and a morphism f of K-varieties from A^n to A^n with an isolated zero at the origin,Eisenbud-Levine and Khimshiashvili defined an algebraic analog (the EKL-degree) for f of the notion of the local Brouwer degree from topology. The EKL-degree takes values in the Grothendieck-Witt ring of K. This talk will discuss a computation of the EKL degree of morphisms induced by quotients under actions of Weyl groups.
(Based on joint work with A. Swaminathan and D. Tseng)
November 2: Charles Godfrey (Pacific Northwest National Laboratory)
Location: REC 302
Title: Higher direct images of ideal sheaves of snc divisors
Abstract: We prove invariance results for the cohomology groups of ideal sheaves of simple normal crossing divisors under (a restricted class of) birational morphisms of pairs in arbitrary characteristic. As an application, we extend a foundational result in the theory of rational pairs that was previously known only in characteristic 0.
November 9: Lisa Marquand (Stony Brook)
Location: REC 302
Title: Symplectic Birational Involutions of manifolds of OG10 type.
Abstract: Compact Hyperkähler manifolds are one of the building blocks of Kähler manifolds with trivial first chern class, but very few examples are known. One strategy for potentially finding new examples is to look at finite groups of symplectic automorphisms of the known examples, and study the fixed loci or quotient. In this talk, we will obtain a classification of birational symplectic involutions of manifolds of OG10 type. We do this from two vantage points: firstly following classical techniques relating birational transformations to automorphisms of the Leech lattice. Secondly, we look at involutions that are obtained from cubic fourfolds via the compactified intermediate Jacobian construction. In this way, we obtain new involutions that could potentially give rise to new holomorphic symplectic varieties. If time permits, we will mention ongoing work to identify the fixed loci in one of these examples.
November 16: Haoyang Guo (Max Planck Institute for Mathematics, Bonn)
Location: REC 302 and Zoom (https://purdue-edu.zoom.us/j/97786675764?pwd=ZUxkQzFsck5IaFBBZnhiOVNTZys4UT09)
Title: A prismatic approach to Fontaine's C_crys conjecture
Abstract: Let X be a smooth proper scheme over a p-adic ring of integers. Motivated by de Rham comparison in complex geometry, Grothendieck asked if the étale cohomology of its generic fiber is comparable to the crystalline cohomology of its special fiber. This was later formulated by Fontaine and is known as C_crys conjecture. In this talk, we give a prismatic proof of the conjecture, for general coefficients, in the relative setting, and allowing ramified base rings. This is a joint work with Emanuel Reinecke.
November 30: Yuchen Liu (Northwestern University)
Location: REC 302
Title: Moduli of log Calabi-Yau pairs
Abstract: 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 Calabi-Yau varieties remains less well understood. I will discuss a new approach to this problem in the case of log Calabi-Yau pairs (X,D), where X is a Fano variety and D is an anticanonical Q-divisor, in which we consider all semi-log-canonical degenerations. One challenge of this approach is that the moduli stack can be unbounded. Nevertheless, if we consider log Calabi-Yau pairs as degenerations of P^2 with plane curves, we show that there exists a projective good moduli space despite the unboundedness. This is ongoing joint work with K. Ascher, D. Bejleri, H. Blum, K. DeVleming, G. Inchiostro, and X. Wang.
December 7: Feng Hao (KU Leuven)
Location: Zoom (https://purdue-edu.zoom.us/j/97786675764?pwd=ZUxkQzFsck5IaFBBZnhiOVNTZys4UT09)
Title: Holomorphic 1-forms on varieties of Kodaira codimension one
Abstract: In principle, holomorphic 1-forms of irregular smooth projective variety X encode much information of the topology and birational geometry of X. In this talk, I will first give a survey on some works and problems about how the zeros of holomorphic 1-forms affect the birational property and topology of smooth projective varieties. Then I will discuss some recent results about holomorphic 1-forms on smooth projective varieties of Kodaira codimension one ($\kappa(X)=\dim X-1$).