Humboldt-Universität zu Berlin
Mathem.-Naturwissenschaftliche Fakultät
Institut für Mathematik
Wintersemester 2025-2026
Time: Wednesday 13:15 - 14:45
Room: 3.007 John von Neumann-Haus
Humboldt Arithmetic Geometry Seminar
Seminar: Algebraic Geometry an der FU
| Time | Speaker | |
|---|---|---|
| 22.10.2025, 13:15 - 14: 45 | Jinhyung Park (KAIST Daejeon) | |
| Title: Determinantal ideals of secant varieties | ||
| Abstract: We show that the homogeneous ideals of secant varieties of smooth projective curves and surfaces in sufficiently ample embeddings are determinantally presented. The same result holds for the first secant varieties of arbitrary smooth projective varieties in sufficiently ample embeddings. This completely settles a conjecture of Eisenbud-Koh-Stillman for curves and partially resolves a conjecture of Sidman-Smith in higher dimensions. To establish the results, we employ the geometry of Hilbert schemes of points. Based on our method, we also prove that the homogeneous ideals of arbitrary projective schemes in sufficiently ample embeddings are generated by quadrics of rank three, confirming a conjecture of Han-Lee-Moon-Park. This is joint work with Daniele Agostini. | ||
| 29.10.2025, 13:15 - 14:45 | Pawel Borowka (Jagiellonian University) | |
| Title: Some questions about abelian surfaces | ||
| Abstract: At the end of 19th century, Humbert characterised the locus of principally polarised abelian surfaces that contain elliptic curves. This locus has infinitely many irreducible components, called Humbert surfaces. The components can be indexed by natural numbers that are exponents of complementary elliptic curves. We would like to describe irreducible components for non-principally polarised abelian surfaces. In the talk, after a brief overview of principally polarised case, I will show some results based on a joint paper with R. Auffarth. I will restate some questions posed in the article and if time permits I will talk about a possible way to answer them. | ||
| 5.11.2025, 13:15 - 14:45 | Daebeom Choi (University of Pennsylvania) | |
| Title: Extremal effective curves and non-semiample line bundles on the moduli space of curves. | ||
| Abstract: In this work, we develop a new method for establishing extremality in the closed cone of curves on the moduli space of curves and determine the extremality of many boundary 1-strata. As a consequence, by using a general criterion for non-semiampleness that extends Keel's argument, we demonstrate that a substantial portion of the cone of nef divisors on the moduli space of pointed curves is not semiample. As an application, we construct the first explicit example of a non-contractible extremal ray of the closed cone of effective curves on the moduli space of n-pointed curves of genus 3. Moreover, we show that this extremal ray is contractible in characteristic p. Our method relies on two main ingredients: (1) the construction of a new collection of nef divisors, and (2) the identification of a tractable inductive structure on the Picard group, arising from Knudsen's construction of the moduli space. | ||
| 10.11.2025, 13:15 - 14:45 ROOM 3.011 | Rahul Pandharipande (ETH Zürich) | |
| Title: On points on lines and lines on planes | ||
| Abstract: I will discuss aspects of the virtual geometry of the moduli space of n lines on the projective plane. The various points of view on the geometry (KSBA, log GW theory, Chow quotients, matroidal decompositions, and so on ...) all contribute to the study. Beyond the construction of the virtual class, I will present an outline of a theory of descendent integration. The talk represents joint work in progress with Dan Abramovich and Dhruv Ranganathan. | ||
| 12.11.2025, 13:15 - 14:45 | Nathan Pflueger (Amherst College) | |
| Title: Brill–Noether theory of twice-marked curves via integer permutations | ||
| Abstract: Classical Brill–Noether theory concerns the following question: if \(C\) is a general curve of genus \(g\), and we fix two positive integers \(a\) and \(b\), what is the geometry of the locus of line bundles \(L\) on \(C\) such that \(h^0(L) = a\) and \(h^1(L) = b\)? I will describe a refinement of this question for twice-marked curves \((C,p,q)\), in which we specify \(h^0\) and \(h^1\) of not just \(L\), but of all twists \(L(ap+bq)\). I will refer to these as transmission loci. Transmission loci provide a convenient tool in degeneration arguments, and their geometry contains rich combinatorics arising from the theory of integer permutations. I will discuss some analogs of classical Brill–Noether theory for transmission loci, and some open problems. | ||
| 26.11.2025, 13:15 - 14:45 | Margherita Lelli-Chiesa (Universita Roma Tre) | |
| Title: Wahl’s extendability theorem via multiple curves | ||
| Abstract: A projective variety in an \(n\)-dimensional projective space is called extendable if it can be realized as a hyperplane section in a projective space of dimension \(n+1\) of a variety which is not a cone. In the 90s Wahl reduced the question about extendability of canonical curves to a vanishing statement; his proof deeply realizes on the deformation theory of the affine cone. This result was essential for the characterization of hyperplane sections of K3 surfaces accomplished by Arbarello-Bruno-Sernesi some twenty years later. In this talk I will present an alternative proof of Wahl's theorem based on multiple curves, that is, non-reduced structures over smooth irreducible curves. This is joint work with Arbarello and Bruno. | ||
| 3.12.2025, 13:15 - 14:45 | Jan Lange (Hannover)) | |
| Title: Torsion order and irrationality of complete intersections | ||
| Abstract: We present the ideas of joint work with Stefan Schreieder, which shows that a very general Fano hypersurface of degree \(d \geq 4\) and dimension at most \((d+1)2^{d-4}\) over a field of characteristic different from 2 is not (retract) rational; thus generalizing results of Schreieder and Moe. Building on these ideas, we extend the above bound to complete intersections improving earlier results of Chatzistamatiou--Levine and Nicaise--Ottem. This is joint work with Guoyun Zhang. | ||
| 10.12.2025, 13:15 - 14:45 | Sandro Verra (Roma) | |
| Title: On cubic hypersurfaces and their nets of polar quadrics | ||
| Abstract: A complex cubic fourfold \(X\) is said to be non special if its Neron-Severi group of codimension 2 cycles is generated by the squared hyperplane class. In the moduli space of cubics the locus of special points splits in a countable union of irreducible divisors. In this talk an explicit example of a nonspecial divisor, with special geometric meaning, is constructed. This adds up to other concrete examples constructed by Auel - Addington and Ranestad - Voisin. It is the moduli space of cubic fourfolds \(X\) whose polar linear system admits a net of quadrics with discriminant a 10-nodal, integral plane sextic. The main result presented is the content of the recent Doctoral Thesis of Elena Sammarco, advised by the speaker. | ||
| 7.01/2026, 13:15 - 14:45 | George Hitching (OSLOMET) | |
| Title: Secant loci of scrolls over curves | ||
| Abstract: The secant loci associated to a linear system \(l\) over a curve C parametrise effective divisors which impose fewer conditions than expected on \(l\). For a rank \(r\) bundle \(E\) and a space of global sections of \(E\), we define and investigate generalised secant loci, which are determinantal loci on Quot schemes of torsion quotients of \(E\). We extend the Abel-Jacobi map to the context of Quot schemes, and examine the relation between smoothness of generalised secant loci and their associated Brill-Noether loci. In one case, we indicate how formulas of Oprea-Pandharipande and Stark can be used to enumerate the generalised secant locus when it has and attains expected dimension zero. | ||