Humboldt-Universität zu Berlin
Mathem.-Naturwissenschaftliche Fakultät
Institut für Mathematik

Online Forschungsseminar "Algebraische Geometrie"

Wintersemester 2020/21

Das Forschungsseminar findet mittwochs in der Zeit von 15:00 - 16:00 Uhr in Zoom statt.

Registration is required here

Seminar: Algebraic Geometry an der FU

04.11.2020 Zhuang He (Humboldt-Universität zu Berlin)
Title: Birational geometry of blow-ups of the projective space along linear subspaces and automorphisms of Kummer surfaces

Abstract: We review some results on the cones of effective divisors and the Mori Dream Space property of \(\overline{M}_{0,n}\) and the blow-ups of projective spaces along linear subspaces. In a joint work with Lei Yang, we showed that the blow-up \(X\) of \(\mathbb{P}^3\) along 6 very general points and 15 lines has infinitely many extremal effective divisors, by constructing a pseudo-automorphism of infinite order on \(X\), which lifts Keum's infinite-order automorphism of a general Kummer surface. With this we proved that the blow-up of \(\overline{M}_{0,n}\) at a very general point has a non-polyhedral effective cone for \(n\) at least 7. We will discuss some ongoing work concerning the pseudo-automorphism group of \(X\), with ideas towards a birational version of the Morrison-Kawamata cone conjecture for this space.
11.11.2020 Thomas Krämer (Humboldt-Universität zu Berlin)
Title: Semicontinuity of Gauss maps and the Schottky problem (slides)

Abstract: We show that the degree of the Gauss map for subvarieties of abelian varieties is semicontinuous in families, and we discuss the loci where it jumps. In the case of theta divisors this gives a finite stratification of the moduli space of ppav's whose strata include the locus of Jacobians and the Prym locus. This is joint work with Giulio Codogni.
18.11.2020 (16:00 - 17:00) Michael Kemeny (University of Wisconsin-Madison)
Title: Minimal rank generators for syzygies of canonical curves

Abstract: The classical Noether-Petri-Babbage Theorem states that a canonical curve is an intersection of quadrics (with a few exceptions). Andreotti-Mayer asked in 1967 whether one could cut the curve out in projective space using only quadrics of rank four or less. This question, which has important implications for the Torelli type problems, was answered in the affirmative by Mark Green in 1984. We will define a notion of rank which makes sense for all linear syzygies of a projective variety, generalizing the rank of the quadrics in its ideal, and then state a generalization of Andreotti-Mayer's conjecture for all linear syzygy spaces. We will also show how to prove the resulting conjecture for the last linear strand of a general canonical curve.
25.11.2020 (16:00 - 17:00) Nicola Tarasca (Virginia Commonwealth University)
Title: Motivic classes of degeneracy loci and pointed Brill-Noether varieties

Abstract: Motivic Chern and Hirzebruch classes are polynomials with K-theory and homology classes as coefficients, which specialize to Chern-Schwartz-MacPherson classes, K-theory classes, and Cappell-Shaneson L-classes. I will present formulas to compute the motivic Chern and Hirzebruch classes of Grassmannian and vexillary degeneracy loci. As an application, I will show how to deduce the Hirzebruch χy-genus of classical and one-pointed Brill-Noether varieties, and therefore their topological Euler characteristic, holomorphic Euler characteristic, and signature. Joint work with Dave Anderson and Linda Chen.
02.12.2020 Gaëtan Borot (Humboldt-Universität zu Berlin)
Title: ELSV and topological recursion for double Hurwitz numbers

Abstract: Hurwitz numbers enumerate of branched cover over \(\mathbb{P}^1\) with various conditions on their ramification. For simple Hurwitz numbers: Ekedahl, Lando, Shapiro and Vainshtein established in 1999 a formula for simple Hurwitz numbers (arbitrary ramification above 0, simple branchpoints elsewhere) in terms of Hodge class intersections on the moduli space of curves; based on mirror symmetry ideas from physics, Bouchard-Marino conjectured a topological recursion to compute them, later proved by Eynard, Mulase and Safnuk. Okounkov-Pandharipande linked Hurwitz numbers to Gromov-Witten theory of \(\mathbb{P}^1\) and Toda integrable hierarchy. These three aspects can be used to study other types of Hurwitz problems. I will describe a joint work with Do, Karev, Lewanski and Moskowsky in which we prove topological recursion and an ELSV-like formula for double Hurwitz numbers (arbitrary ramification over 0 and infinity, simple branching elsewhere), and explain consequences for vanishing of Chiodo class integrals, generalizing results of Johnson, Pandharipande and Tseng.
09.12.2020 (16:00 - 17:00) Lawrence Ein (University of Illinois at Chicago)
Title: TBA.

16.12.2020 Yohan Brunebarbe (Université de Bordeaux)
Title: TBA.

23.12.2020 TBA.

30.12.2020 TBA.

06.01.2021 TBA.

13.01.2021 TBA.

20.01.2021 TBA.

27.01.2021 TBA.

03.02.2021 TBA.

10.02.2021 TBA.

17.02.2021 TBA.



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