Abstract: I will construct LG mirrors for the Johnson-Kollár series of anticanonical del Pezzo surfaces in weighted projective 3-spaces. The main feature of these surfaces is that their anticanonical linear system is empty. Thus they fall outside of the range of the known mirror constructions. For each of these surfaces, the LG mirror is a pencil of hyperelliptic curves. I will exhibit the regularised I-function of the surface as a period of the pencil and I will sketch how to construct the pencil starting from a work of Beukers, Cohen, and Mellit on finite hypergeometric functions. This is joint work with Alessio Corti.

Abstract: The talk will revolve around combinatorial aspects Prym varieties, a class of Abelian varieties that occurs in the presence of double covers. Pryms have deep connections with torsion points of Jacobians, bi-tangent lines of curves, and spin structures. As I will explain, problems concerning Pryms may be reduced, via tropical geometry, to combinatorial games on graphs. As a consequence, we obtain new results concerning the geometry of special algebraic curves, and bounds on dimensions of certain Brill–Noether loci.

Abstract: I will discuss an invariant of singularities, Saito's minimal exponent, and its connections with various other invariants of singularities. This is based on joint work with Mihnea Popa.

Abstract: It is well known that, for pointed nodal curves, considering flat and proper families of pairs (X,D) leads to a proper moduli space. Still, while the notion of stable pairs is a higher dimensional analogue of pointed nodal curves, the right definition of a family of stable pairs is far from obvious. In this work, building on an idea of Kollár and the work of Abramovich and Hassett, we give an alternative definition of a family of stable pairs, in the case where the divisor D is reduced. This definition is more amenable to the tools of deformation theory. As an application we produce functorial gluing morphisms on the moduli spaces of surfaces, generalizing the clutching and gluing morphisms that describe the boundary strata of the moduli of curves. This is joint work with D. Bejleri.

Abstract: In a series of papers, Aluffi and Faber computed the degree of the GL3 orbit closure of an arbitrary plane curve. We attempt to generalize this to the equivariant setting by studying how these orbits degenerate, yielding a fairly complete picture in the case of plane quartics. As an enumerative consequence, we will see that a general genus 3 curve appears 510720 times as a 2-plane section of a general quartic threefold. We also hope to survey the relevant literature and will only assume the basics of intersection theory. This is joint work with M. Lee and A. Patel.

Abstract: I substitute the original speaker Kento Fujita, who has to cancel his trip. In this talk, I will survey the development of algebraic geometer’s understanding of K-stability of Fano varieties. K-stability was originally defined by Tian and Donaldson. Later, Fujita and Li found an equivalent valuative criterion to test K-stability. This has significantly improved the connection of K-stability with other branches of higher dimensional geometry, and underlies many recent improvements.

Abstract:By the Torelli theorem, the moduli space [M] of lattice-polarized K3 surfaces is the quotient of a Hermitian symmetric domain by an arithmetic group. In this capacity, it has compactifications such as the Baily-Borel [\overline{M}^{\rm BB}] and toroidal compactifications [\overline{M}^F] which depend on some choice of fan [F] . On the other hand, choosing canonically an ample divisor on every such K3, one can build a compactification [\overline{M}^{\rm slc}] of so-called stable pairs.

I will discuss joint work with V. Alexeev on how one proves that the normalization of [\overline{M}^{\rm slc}] is [\overline{M}^F] for some choice of fan [F] . We will focus on the example of elliptic K3s, polarized either by either the trisection [R^{\rm ram}] of nontrivial -torsion or by [R^{\rm rc}] : The section plus the sum of the singular fibers.

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Abstract: A Milnor square is a bicartesian square in schemes where the bottom arrow is a closed immersion. This captures the idea of "gluing along a closed subscheme." An abstract blowup square is a cartesian square of schemes where one leg is a closed immersion and the other is a birational proper morphism. This is a simple generalization of the squares one obtains by blowups. Given a cohomology theory for schemes (valued in complexes or spectra), it is natural to ask if it converts these squares to homotopy cartesian squares, resulting in some useful long exact sequences. This will be an examples-based talk on very recent progress and revisions made in this old question, including work of Kerz-Strunk-Tamme for K-theory, Bhatt-Mathew on étale cohomology and joint work with Hoyois-Iwasa-Kelly on (higher) Chow groups.

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