Humboldt-Universität zu Berlin
Mathem.-Naturwissenschaftliche Fakultät
Institut für Mathematik
Sommersemester 2024
Time: Wednesday 13:15 - 14:45
Room: 3.007 John von Neumann-Haus
Humboldt Arithmetic Geometry Seminar
Seminar: Algebraic Geometry an der FU
| Time | Room | Speaker |
|---|---|---|
| THURSDAY 18.04.2024, 13:15h | Gregory Pearlstein (Pisa) | |
| Title: KSBA stable limits associated to quasi-homogeneous surface singularities | ||
| Abstract: Smooth minimal surfaces of general type with \(K^2=1\), \(p_g=2\), and \(q=0\) constitute a fundamental example in the geography of algebraic surfaces. The moduli space of their canonical models admits a modular compactification \(M\) via the minimal model program. In previous work with Patricio Gallardo and Luca Schaffler we constructed eight new irreducible boundary divisors in \(M\) arising from unimodal singularities. In this talk, we will discuss extension of this work to quasi-homogeneous surface singularities. | ||
| 15.05.2024, 13:15h | Nazim Khelifa (IHES) | |
| Title: Density criteria for typical Hodge loci and applications | ||
| Abstract: After recalling the Zilber-Pink paradigm introduced in Hodge theory by Klingler and further developed by Baldi-Klingler-Ullmo, I will present joint work with David Urbanik giving sufficient conditions that ensure that the Hodge locus, i.e. the locus in the base of an integral polarized variation of Hodge structures where the fibers acquire non-generic Hodge tensors, is dense for the complex analytic topology in the base. I will then explain how to relate this result to classical results on Noether-Lefschetz loci. Finally, I will explain how the current knowledge of the Hodge locus can be used to revisit and improve classical bounds on the dimension of the image of period maps, studied among others by Carlson, Griffiths, Kasparian, Mayer and Toledo. | ||
| 22.05.2024, 13:00-14:00h | Alexandru Suciu (Northeastern University) | |
| Title: The Milnor fibrations of hyperplane arrangements | ||
| Abstract: To each multi-arrangement \((A,m)\), there is an associated Milnor fibration of the complement \(M=M(A)\). Although the Betti numbers of the Milnor fiber \(F=F(A,m)\) can be expressed in terms of the jump loci for rank 1 local systems on \(M\), explicit formulas are still lacking in full generality, even for \(b_1(F)\). After introducing these notions and explaining some of the known results, I will consider the "generic" case, in which \(b_1(F)\) is as small as possible. I will describe ways to extract information on the cohomology jump loci, the lower central series quotients, and the Chen ranks of the fundamental group of the Milnor fiber in this situation. | ||
| 22.05.2024, 14:15-15:15h | Marian Aprodu (University of Bucharest) | |
| Title: Resonance, syzygies, and rank-3 Ulrich bundles on the del Pezzo threefold \(V_5\) | ||
| Abstract: This is a joint work with Yeongrak Kim. We investigate a geometric criterion for a smooth curve of genus 14 and degree 18 to be described as the zero locus of a section in an Ulrich bundle of rank 3 on a del Pezzo threefold \(V_5\). The main challenge is to read off the Pfaffian quadrics defining \(V_5\) from geometric properties of the curve. We find that this problem is related to the existence of a special rank-two vector bundle on the curve, with trivial resonance. From an explicit calculation of the Betti table, we also deduce the uniqueness of the del Pezzo threefold. | ||
| 30.05.2024-31.05.2024 | Workshop Curves, abelian varieties, and their moduli at HU Berlin | |
| Curves, abelian varieties, and their moduli | ||
| TUESDAY 04.06.2024, 13:30-14:30h | ROOM 3.006 | Richard Rimanyi (UNC Chapel Hill and Newton Institute Cambridge) |
| Title: 3d mirror symmetry for characteristic classes | ||
| Abstract: In this joint work with Tommaso Botta we study the elliptic characteristic classes called stable envelopes introduced by M. Aganagic and A. Okounkov. Stable envelopes measure singularities, they geometrize quantum group representations, and they can be interpreted as monodromy matrices of certain differential or difference equations. We prove that for a rich class of holomorphic symplectic varieties (called bow varieties) their elliptic stable envelopes display a duality inspired by mirror symmetry in \(d=3\), \(N=4\) quantum field theories. In the key step of our proof, we "resolve" large charge branes to a number of smaller charge branes. This phenomenon turns out to be the geometric counterpart of the algebraic fusion procedure. Along the way we discover more about the rich geometry of bow varieties, such as their Bruhat order and the elliptic Hall algebra structure on their stable envelopes. | ||
| 12.06.2024, 13:15h | Ruijie Yang (HU Berlin) | |
| Title: Minimal exponent of a hypersurface | ||
| Abstract: Recently, the minimal exponent of a hypersurface over complex numbers has been understood as a useful refined invariant of the log canonical threshold. It has found many new applications including deformation of Calabi-Yau 3-folds (Friedman-Laza), higher rational/du Bois singularities (Mustata-Popa) and geometric Schottky problem (Schnell-Yang). However, some basic properties of this invariant remain mysterious. In this talk I will discuss the conjecture of Mustata and Popa on birational characterization of the minimal exponent, which is the main obstruction for the computation in practice. I will explain the heuristic of the Mustata-Popa conjecture from Igusa's work on counting integer solutions of congruence equations and Igusa’s strong monodromy conjecture. Then I will discuss how several ideas from mixed Hodge modules and geometric representation theory can lead to a better understanding of the minimal exponents. This is based on two joint works with Christian Schnell and Dougal Davis, respectively. | ||
| 19.06.2024, 13:15h | Carl Lian (Tufts University) | |
| Title: Complete quasimaps to \(Bl_p(\mathbb{P}^2)\) | ||
| Abstract: We consider the problem of counting curves \(C\) of fixed moduli in a target variety \(X\) passing through the maximal number of points ("Tevelev degrees"). A broad program for obtaining such a count is: (i) Establish a Brill-Noether theorem for maps to \(X\). (ii) Use (i) to construct a compact moduli space \(M\), generically parametrizing maps to \(X\), which witnesses (without excess intersections) the desired count. (iii) Compute integrals on \(M\), e.g., by degeneration methods. Typically, the moduli spaces coming from, e.g., Gromov-Witten theory, are not sufficient in step (ii). We review the case \(X=\mathbb{P}^r\), where this program has been carried out. Here, one may take \(M\) to be the moduli space of complete collineations (relative to the space of linear series on \(C\)), which is an iterated blow-up of a Quot scheme. We then report on work in progress with Alessio Cela on the case where \(X\) is a blow-up of \(\mathbb{P}^2\) at a point. Here, a Brill-Noether statement is given (more generally, for the blow-up of \(\mathbb{P}^r\) at any linear space) by a result of Farkas, and we construct a moduli space \(M\) of "complete quasimaps" to \(X\). Degenerations on this space are made possible by ideas from the previous calculation on \(\mathbb{P}^r\) and from Coskun's geometric Littlewood-Richardson rule. Our construction seems to hint toward a more general theory of complete quasimaps to other targets. | ||