# Stefano Marseglia

## AG Seminar at Utrecht University

Since the end of February 2023, the webpage of the AG Seminar has moved here!

This is the Algebraic Geometry and Arithmetic Geometry seminar and features both research talks, as well as preprint talks presenting papers that appeared recently on arXiv in the algebraic geometry or the number theory sections and are of interest to the seminar participants.

If you wish to participate, contact me at s.marseglia (at) uu.nl.

If you wish to to be added to the mailing list, contact Marta Pieropan at m.pieropan (at) uu.nl.

The seminar takes place on Wednesdays at 13:30 in HFG-409 unless stated otherwise, until the end of Block 3.
In Block 4, starting May 10, the seminar will take place at 11:30 in HFG-409, followed by lunch.

### Upcoming Talks

• March 29
• April 5, (13:30, HFG-409) research talk by Fabian Reede (Hannover) Line bundles on noncommutative algebraic surfaces
Abstract: Let X be a complex algebraic surface. Let A be a (noncommutative) coherent O_X-algebra whose generic stalk is a central division algebra. Then the pair (X,A) can be thought of as a noncommutative algebraic surface. This talk deals with the moduli space of line bundles on (X,A), i.e., of locally free A-modules of rank one, in the cases where X has Kodaira dimension zero or is Fano. It turns out that the moduli space can be compactified by adding torsion-free A-modules of rank one. We will study e.g. the smoothness of this compactification and the deformation theory of the sheaves involved. This is partly joint work with U. Stuhler and with N.Hoffmann.
• April 13, [ this week : NMC ]
• April 18, (13:30-15:30) Minicourse by Carolina Tamborini (UU)
• April 19, (13:30, HFG-409) research talk by Oliver Leigh (Uppsala)
• April 20, (13:30-15:30) Minicourse by Carolina Tamborini (UU)
• April 25, (13:30-15:30) Minicourse by Carolina Tamborini (UU)
• April 26, (13:30-15:30) Minicourse by Carolina Tamborini (UU)
• May 3, (13.30-14.30, HFG 409) Chia-Fu Yu (Academia Sinica, Taiwan) Arithmetic invariants on supersingular Ekedahl-Oort (EO) strata for g=4
Abstract: The supersingular locus is one of main interests in algebraic geometry in characteristic p, and can be described in terms of polarised flag type quotients (PFTQs) in the sense of Li and Oort. The description for g=3 is rather explicit and is exploited by Karemaker and Yobuko and myself for investigating the arithmetic invariants of supersingular abelian threefolds, namely, the endomorphism rings and automorphism groups of them, confirming Oort's conjecture for g=3. In this talk we shall explain a general method for investigating the arithmetic invariants on supersingular EO strata, and report the progress of the joint work with Karemaker for g=4.
• May 10, (11:30-12:30) Thorsten Schimannek
• May 17
• May 24
• May 31
• June 7
• June 14
• June 21
• June 28

#### Past Talks

2023

• March 22, (13:30, HFG-409) research talk by Sara Mehidi (Bordeaux) Extending torsors via log schemes
Abstract: We give here an approach of the problem of extending torsors defined on the generic fiber of a family of curves. The question is to extend each of the structural group and the total space of the torsor above the family. This problem has been studied by many researchers, starting by the first ideas of Grothendieck who solved the case of a constant group of order prime to the residual characteristic. When we are interested in algebraic varieties from an arithmetic point of view, it is natural to consider torsors under a finite flat group that is not necessarily constant: we talk about fppf torsors. In fact, we know from the literature that there are cases where the problem does not have a solution in this setting.
So the idea is to look for a solution in a larger category, namely the category of logarithmic torsors. We will show in particular that the existence of such an extension amounts to extending group functors and morphisms between them. Then, we will compute the obstruction for the extended log torsor to lift into an fppf one. Finally, we give an example of a computation of an extension of a torsor over a given hyperelliptic curve as an application of our results.
• March 15, (13:30, HFG-409) research talk by Henry Liu (Oxford): Multiplicative vertex algebras and wall-crossing in equivariant K-theory
Abstract: I will give an overview of recent progress in wall-crossing for equivariant K-theoretic invariants of moduli of sheaves, based on a new framework of Joyce. A multiplicative version of vertex algebras plays a central role in this and related stories. I will give some applications to refined Vafa-Witten theory, the 3-fold DT/PT correspondence, and, if time permits, also some speculation about modularity and S-duality.
• March 1, (13:30, HFG-409) research talk by Stefano Marseglia (UU): Cohen-Macaulay type of endomorphism rings of abelian varieties over finite fields
Abstract: In this talk, we will speak about the (Cohen-Macaulay) type of the endomorphism ring of abelian varieties over a finite field with commutative endomorphism algebra. We will exhibit a condition on the type of End(A) implying that A cannot be isomorphic to its dual. In particular, such an A cannot be principally polarised or a Jacobian. This is partly joint work with Caleb Springer.
• February 15, (13:30, BBG-069) research talk by Andreas Braun (Durham): Hodge classes on Calabi-Yau fourfolds
Abstract: A crucial question in string theory concerns its set of solutions, particularly those giving rise to an effective four-dimensional description. The most general such solutions are found by specifying a Calabi-Yau fourfold equipped with an elliptic fibration, together with a Hodge class. Without assuming any background I will briefly explain how these objects arise in physics, and which constraints they need to obey. After formulating some of the central questions and conjectures that arise in this context, I will discuss some recent progress.
• February 8, (13:30, HFG-409) research talk by Hyeonjun Park (KIAS): Cosection localization via derived algebraic geometry
Abstract: Cosection localization is one of the most powerful tools in virtual enumerative geometry. In this talk, we revisit cosection localization from the perspective of derived algebraic geometry. I will explain derived reduction by (-1)-shifted closed 1-forms and localization through homotopical intersection theory. I will also provide an intrinsic description of cosection-localized virtual cycles using (-2)-shifted symplectic structures. This is based on joint works with Younghan Bae and Martijn Kool, with Dhyan Aranha, Adeel Khan, Alexei Latyntsev, and Charanya Ravi, and with Young-Hoon Kiem.
• February 1, (11:00, HFG-409) research talk by Olivier de Gaay Fortman (Hannover): Real moduli spaces, unitary Shimura varieties and non-arithmetic lattices
Abstract: Hodge theory can sometimes be used to identify a moduli space of complex varieties with a complex ball quotient, or an open subset of such a space. I will explain that similar things happen for moduli of real varieties. Real moduli spaces of smooth varieties are often not connected, however - to get a connected moduli space one is led to allow some singularities. It turns out that, very similar to the way in which the connected components of the space of smooth varieties embed into the larger moduli space, real arithmetic ball quotients can be glued together to form a large real ball quotient. Unitary Shimura varieties provide the right framework for this glueing procedure. I will explain how this works, constructing non-arithmetic lattices in PO(n,1) for every n.
• January 18, (10:00, HFG-409) research talk by Mick van Vliet (UU): Tame geometry and Hodge theory
Abstract: Tame geometry, made precise by the concept of o-minimal structures, has recently led to some interesting developments in algebraic geometry. In the first half of this talk I will motivate and explain the definition of o-minimal structures, and review some remarkable theorems that hold in the resulting framework of tame geometry. In the second half of the talk, based on work of Bakker, Klingler, and Tsimerman (1810.04801), I will give an overview of a recent application of tame geometry to Hodge theory.
• January 11, (11:00, HFG-409) research talk by Lars Halvard Halle (University of Bologna): Degenerations of Hilbert schemes and relative VGIT
Abstract: This talk will be a report on joint work with K. Hulek and Z. Zhang. First I will explain how some central results in VGIT can be extended to a relative setting. After this, I will discuss an application of relative VGIT to the study of certain degenerations of Hilbert schemes of points.

2022

• December 7, (HFG-409) research talk by Leo Herr (Leiden University): The Rhizomic Topology
Abstract: What is a sheaf on a log scheme X? If we take the ordinary etale topology, we ignore the log structure. Taking the log étale topology, even the structure "sheaf" O_X is not a sheaf! The same goes for M_X, \overline M_X. We introduce a new "rhizomic" topology on log schemes coarser than the log etale topology. Will this be enough?
• November 30, (HFG-409) research talk by Navid Nabijou (QMU London): Roots and logs in the enumerative forest
Abstract: Logarithmic and orbifold structures provide two different paths to the enumeration of curves with fixed tangencies to a normal crossings divisor. Simple examples demonstrate that the resulting systems of invariants differ, but a more structural explanation of this defect has remained elusive. I will discuss joint work with Luca Battistella and Dhruv Ranganathan, in which we identify birational invariance as the key property distinguishing the two theories. The logarithmic theory is stable under strata blowups of the target, while the orbifold theory is not. By identifying a suitable system of “slope-sensitive” blowups, we define a “limit" orbifold theory and prove that it coincides with the logarithmic theory. Our proof hinges on a technique – rank reduction – for reducing questions about normal crossings divisors to questions about smooth divisors, where the situation is much-better understood.
• November 23, (HFG-409) research talk by Mar Curco Iranzo (UU): Generalised Jacobians of modular curves and their Q-rational torsion
Abstract: The Jacobian J0(N) of the modular curve X0(N) has received much attention within arithmetic geometry for its relation with cusp forms and elliptic curves. In particular, the group of Q-rational points on X0(N) controls the cyclic N-isogenies of elliptic curves. A conjecture of Ogg predicted that, for N prime, the torsion of this group comes all from the cusps. The statement was proved by Mazur and later generalised to arbitrary level N into what we call generalised Ogg’s conjecture. Consider now the generalised Jacobian J0(N)m with respect to a modulus m. This algebraic group also seems to be related to the arithmetic of X0(N) through the theory of modular forms. In the talk we will present new results that compute the Q-rational torsion of J0(N) for N an odd integer with respect to a cuspidal modulus m. These generalise previous results of Yamazaki, Yang and Wei. In doing so, we will also discuss how our results relate to generalised Ogg’s conjecture.
• November 16, (HFG-409) research talk by Francesca Carocci (EPFL): BPS invariant from non Archimedean integrals
Abstract: We consider moduli spaces of one-dimensional semistable sheaves on del Pezzo and K3 surfaces supported on ample curve classes. Working over a non-archimedean local field F, we define a natural measure on the F-points of such moduli spaces. We prove that the integral of a certain naturally defined gerbe on the moduli spaces with respect to this measure is independent of the Euler characteristic. Analogous statements hold for (meromorphic or not) Higgs bundles. Recent results of Maulik-Shen and Kinjo-Coseki imply that these integrals compute the BPS invariants for the del Pezzo case and for Higgs bundles. This is a joint work with Giulio Orecchia and Dimitri Wyss.
• October 26, (HFG-409) research talk by Reinier Schmiermann: On Classifying Continuous Constraint Satisfaction Problems
Abstract: The computational complexity class of the existential theory of the reals contains problems which can be reduced to checking whether a system of polynomial equations has a solution over the real numbers. The complexity of a lot of problems in computational geometry turns out to be captured by this class (they are complete for this class). These completeness proofs often use the completeness of a specific Continuous Constraint Satisfaction Problem (CCSP) as an intermediate step. We attempt to give a more systematic analysis of the computational complexity of these CCSPs, and show that a large class of CCSPs is complete for the existential theory of the reals. In this talk, I will give a introduction to computational complexity, the existential theory of the reals, and CCSPs. Then I will state our results, and give a sketch of the proof. This talk is based on joint work with Tillmann Miltzow.
• October 19, (HFG-409) Carel Faber, preprint talk on "On the Chow and cohomology rings of moduli spaces of stable curves" by Canning and Larson, arXiv:2208.02357
• October 12 (HFG-409), research talk by Remy van Dobben de Bruyn: A variety that cannot be dominated by one that lifts
Abstract: The recent proofs of the Tate conjecture for K3 surfaces over finite fields start by lifting the surface to characteristic 0. Serre showed in the sixties that not every variety can be lifted, but the question whether every motive lifts to characteristic 0 is open. We give a negative answer to a geometric version of this question, by constructing a smooth projective variety that cannot be dominated by a smooth projective variety that lifts to characteristic 0.
• October 5, research talk by Dusan Dragutinovic: Computing binary curves of genus five
Abstract. In this talk, we will present algorithms used to determine, up to isomorphism over $\F_2$, all genus five curves defined over $\F_2$ (together with the sizes of their $\F_2$-automorphism groups). Furthermore, we will discuss the outcome considering the Newton polygons of computed curves and mention the obtained stack count $|\mathcal{M}_5(\F_2)|$.
• September 21, Boaz Moerman, preprint talk on "Weak approximation and the Hilbert property for Campana points" by Nakahara and Streeter, arXiv:2010.12555
• July 6 (11:00 - 12:00, HFG-611), research talk by Valentijn Karemaker
• June 28 (11:00 - 13:00, HFG-611), research talk by Pol van Hoften
• June 22 (11:15, HFG-610) Carolina Tamborini, "Punctual characterization of the unitary flat bundle of weight 1 PVHS and application to families of curves" by González-Alonso and Torelli, arXiv:2101.03153
• June 8 (HFG-610), Wilberd van der Kallen, "Frobenius Splittings", arXiv:1208.3100
• May 31 (HFG-610), Dirk van Bree, "When are two HKR isomorphisms equal?" by Huang, arXiv:2205.04439
• May 25 (HGF-610), Marta Pieropan, "Heights on stacks and a generalized Batyrev-Manin-Malle conjecture" by Ellenberg, Satriano and Zureick-Brown, arXiv:2106.11340
• May 9 (Duistermaat), Marta Pieropan, "Global Frobenius liftability I & II" by Achinger, Witaszek and Zdanowicz, arXiv:1708.03777 and arXiv:2102.02788
• March 7, Sebastián Carrillo Santana, "Values of zeta-one functions at positive even integers" by Kobayashi and Sasaki, arXiv:2202.11835
• Feb. 28, Carolina Tamborini, "The Coleman-Oort conjecture: reduction to three key cases" by Moonen, arXiv:2201.11971
• Feb. 14, Boaz Moerman, "Tamagawa measures on universal torsors and points of bounded height on Fano varieties" by Salberger, article, and "Compter des points d'une variété torique" by de la Bretèche, article
• Feb. 7, Sergej Monavari, "On the motive of the Quot scheme of finite quotients of a locally free sheaf" by Ricolfi, arXiv:1907.08123
• Jan. 24, Dirk van Bree, "Using the internal language of toposes in algebraic geometry" by Blechschmidt, arXiv:2111.03685

2021

• Nov. 29 (HFG-611), Reinier Schmiermann, "Components and singularities of Quot schemes and varieties of commuting matrices" by Jelisiejew and Šivic, arXiv:2106.13137
• Nov. 22 (KBG-Atlas), Dusan Dragutinovic, "The existence of supersingular curves of genus 4 in arbitrary characteristic" by Kudo, Harashita and Senda, arXiv:1903.08095
• Nov. 8, Dirk van Bree, "Unramified division algebras do not always contain Azumaya maximal orders" by Antieau and Williams, arXiv:1209.2216
• Nov. 1, Stefano Marseglia, "On matrices of endomorphisms of abelian varieties" by Zarhin, arXiv:2002.00290, and "Lattices in Tate modules" by Poonen and Rybakov, arXiv:2107.06363
• Oct. 11, Carel Faber, "A non-hyperelliptic curve with torsion Ceresa class" by Beauville, arXiv:2105.07160, and "A non-hyperelliptic curve with torsion Ceresa cycle modulo algebraic equivalence" by Beauville and Schoen, arXiv:2106.08390
• Oct. 4, Marta Pieropan, "Sums of four squareful numbers" by Shute, arXiv:2104.06966