## AG Seminar

This is and 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 Marta Pieropan m.pieropan@uu.nl to be added to the mailing list.

The seminar takes place on Wednesdays at 11:00 in HFG-409 unless stated otherwise.

### Talks

Upcoming

• 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?
• January 11, (HFG-409) research talk by Lars Halvard Halle (University of Bologna)
• January 18, (10:00, HFG-409) research talk by Mick van Vliet (UU)
• February 1, (HFG-409) research talk by Olivier de Gaay Fortman (Leibniz University Hannover)
• February 8, (13:30, HFG-409)research talk by Hyeonjun Park (KIAS)
• March 15, (13:30, HFG-409) research talk by Henry Liu (Oxford)
• March 22, (13:30, HFG-409) research talk by Sara Mehidi (Bordeaux)

Past

• 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