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

 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 (notes) 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: Singularities and syzygies of secant varieties of curves Abstract: Let $$X$$ be a smooth curve of genus $$g$$ and $$L$$ be a line bundle of degree at least $$2g+2k+p+1$$. We show that the $$k$$-th secant variety of the pair $$(X,L)$$ satisfies property $$N_{k+2,p}$$. It means that the homogeneous ideal is generated by forms of degree $$k+2$$ and the next $$p$$ steps of the minimal resolutions are given by matrices of linear forms. Furthermore, is a normal projective Cohen Macaulay variety with Du Bois singularities. 16.12.2020 Yohan Brunebarbe (Université de Bordeaux) Title: Increasing hyperbolicity of varieties supporting a variation of Hodge structures with level structures Abstract: Looking at the finite étale congruence covers $$X(p)$$ of a complex algebraic variety $$X$$ equipped with a variation of integral polarized Hodge structures whose period map is quasi-finite, I will explain why both the minimal gonality among all curves contained in $$X(p)$$ and the minimal volume among all subvarieties of $$X(p)$$ tend to infinity with $$p$$. This applies for example to Shimura varieties, moduli spaces of curves, moduli spaces of abelian varieties, moduli spaces of Calabi-Yau varieties, and can be made effective in many cases. I will also discuss some related results on the distribution of entire curves in the compactifications of the $$X(p)$$'s obtained in joint work with Damian Brotbek. 06.01.2021 Leonardo Lerer (Université Paris-Saclay) Title: Cohomology jump loci for singular varieties Abstract: Let $$X$$ be a topological space homotopy equivalent to a finite CW-complex. The cohomology jump loci are certain subvarieties of the Betti moduli space of $$X$$ that parametrize local systems satisfying a dimension condition on their cohomology. When $$X$$ is a complex algebraic manifold, the study of these subvarieties has a rich history and their geometry, for local systems of rank one, is well understood. In this talk, I will present some results on cohomology jump loci in the case when $$X$$ is no longer assumed to be smooth. 13.01.2021 (15:00 - 16:00) Ignacio Barros (Université Paris-Saclay) Title: On the irrationality of moduli spaces of K3 surfaces Abstract: I will report on recent joint work with D. Agostini and K.-W. Lai, where we study how the degrees of irrationality of the moduli space of polarized K3 surfaces grow with respect to the genus $$g$$. We prove that, for a series of infinitely many genera, the irrationality is bounded by the Fourier coefficients of certain modular forms of weight 11, and thus grow at most polynomially, in terms of $$g$$. Our proof relies on results of Borcherds on Heegner divisors together with results of Hassett and Debarre-Iliev-Manivel on special cubic fourfolds and Gushel-Mukai fourfolds. (16:15 - 17:15) Ruijie Yang (Stony Brook University) Title: Decomposition theorem for semisimple local systems Abstract: In complex algebraic geometry, the decomposition theorem asserts that semisimple geometric objects remain semisimple after taking direct images under proper algebraic maps. This was conjectured by Kashiwara and is proved by Mochizuki and Sabbah in a series of long papers via harmonic analysis and D-modules. In this talk, I would like to explain a simpler proof in the case of semisimple local systems using a more geometric approach adapting De Cataldo-Migliorini. As a byproduct, we also recover a weak form of Saito's decomposition theorem for variation of Hodge structures. Joint work in progress with Chuanhao Wei. 20.01.2021 (15:00 - 16:00) Frederik Benirschke (Stony Brook University) Title: Boundary of linear subvarieties Abstract: Strata of differentials are moduli spaces of differential forms on Riemann surfaces with prescribed multiplicity of zeros and poles. Strata have natural coordinates given by the periods of the differential. We are interested in a special class of subvarieties of strata, so-called linear subvarieties. These are algebraic subvarieties which are locally given by linear equations among the periods. Examples of linear varieties arise from both algebraic geometry, for example as Hurwitz Spaces, as well as Teichmüller theory. Recently a compactification of strata, the moduli space of multi-scale differentials, has been constructed by Bainbridge-Chen-Gendron-Grushevsky-Möller. We study the closure of linear subvarieties in the moduli space of multi-scale differentials. Our main result is that the boundary of a linear subvariety is again given by linear equations among periods. Time permitting, we show how our results can be used to construct a compactification of Hurwitz spaces different from admissible covers and how this can be used for class computations. (16:15 - 17:15) Yeuk Hay Joshua Lam (Harvard University) Title: Calabi-Yau varieties and Shimura varieties Abstract: I will discuss the Attractor Conjecture for Calabi-Yau varieties, which was formulated by Moore in the nineties; in particular I will try to highlight the difference between Calabi-Yau varieties with and without "Shimura moduli". In the Shimura case, I show that the conjecture holds and gives rise to an explicit parametrization of CM points on certain Shimura varieties, with relations to arithmetic invariant theory; in the case without Shimura moduli, I'll present counterexamples to the conjecture using unlikely intersection theory. Part of this is joint work with Arnav Tripathy. 27.01.2021 Giacomo Mezzedimi (Universität Hannover) Title: The Kodaira dimension of some moduli spaces of elliptic K3 surfaces Abstract: Let $$\mathcal{M}_{2k}$$ denote the moduli space of $$U\oplus \langle -2k\rangle$$-polarized K3 surfaces. Geometrically, the K3 surfaces in $$\mathcal{M}_{2k}$$ are elliptic and contain an extra curve class, depending on $$k\ge 1$$. I will report on a joint work with M. Fortuna and M. Hoff, in which we compute the Kodaira dimension of $$\mathcal{M}_{2k}$$ for almost all $$k$$: more precisely, we show that it is of general type if $$k\ge 220$$ and unirational if $$k\le 50$$, $$k\notin \{11,35,42,48\}$$. After introducing the general problem, I will compare the strategies used to obtain both results. I will then show some examples arising from explicit geometric constructions. 03.02.2021 No seminar 10.02.2021 Dawei Chen (Boston College) Title: Connected components of the strata of $$k$$-differentials Abstract: $$k$$-differentials on Riemann surfaces are sections of the $$k$$-th power of the canonical bundle. The moduli space of $$k$$-differentials can be stratified according to the multiplicities of zeros and poles of $$k$$-differentials. While these strata are smooth, some of them can be disconnected. In this talk I will review known results and open problems regarding the classification of their connected components, with a focus on geometric structures that can help distinguish different components. This is joint with Quentin Gendron. 17.02.2021 Claire Voisin (College de France) Title: Schiffer variations of hypersurfaces and the generic Torelli theorem Abstract: The generic Torelli theorem for hypersurfaces of degree $$d$$ and dimension $$n-1$$ was proved by Donagi in the 90's. It works under the assumption that $$d$$ is at least 7 and $$d$$ does not divide $$n+1$$, which in particular excludes the Calabi-Yau case in all dimensions. We prove that the generic Torelli theorem for hypersurfaces holds with finitely many exceptions. A key tool is the notion of Schiffer variation of a hypersurface and how to characterize them by looking at how varies the variation of Hodge structure along them. This is thus a second order argument in the theory of variations of Hodge structures. 24.02.2021 No seminar, rescheduled for next week 03.03.2021 Carl Lian (Humboldt-Universität zu Berlin) Title: Non-tautological and H-tautological Hurwitz cycles Abstract: We will explain the construction of a large family of new non-tautological algebraic cycles on moduli spaces of curves coming from Hurwitz spaces. Namely, the locus of stable curves of sufficiently large genus admitting a degree $$d$$ cover of a curve of genus $$h>0$$ is non-tautological, with appropriate marked points added and subject to the non-vanishing of the $$d$$-th Fourier coefficient a certain modular form. This builds on examples of Graber-Pandharipande and van Zelm in the case $$(d,h)=(2,1)$$. Time-permitting, we will discuss how these cycles fit instead into a larger theory of "H-tautological" classes on moduli spaces of admissible Galois covers of curves.

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