Dutch Online Algebraic Geometry Seminar, Spring 2021

The seminar meetings will take place on Fridays (bi-weekly) at 3pm, via zoom. The organisation of the session rotates. Before the talks there is an opportunity to meet informally via gather.town. Announcements with the required coordinates and passwords are sent to the mailing list for the seminar. If you wish to receive these announcements, please subscribe to this mailing list, which can be done here. If you experience any problems with this, please send an email to Ben Moonen, b.moonen =at= science.ru.nl

Speaker: **Remy van Dobben de Bruyn** (Princeton and IAS)

Title: **Unbounded negativity on rational surfaces in positive characteristic**

Abstract: The bounded negativity conjecture asks whether the self-intersections of irreducible curves on a (fixed) surface are bounded from below. It is known to fail for surfaces of general type in positive characteristic, and the rational and characteristic 0 cases have been heavily studied recently. In joint work with Raymond Cheng, we show that the conjecture fails even for rational surfaces in positive characteristic.

Speaker: **Stefan Schreieder** (Leibnitz Universität Hannover)

Title: **Refined unramified homology of schemes**

Abstract: We introduce refined unramified (co-)homology of algebraic schemes and prove several comparison results with cycle groups. Over the complex numbers, our approach simplifies and generalizes to cycles of arbitrary codimensions on possibly singular schemes, previous results of Bloch—Ogus, Colliot-Thélène—Voisin, Voisin, and Ma. Our approach has several applications. For instance, it allows to produce the first example of a smooth complex projective variety whose Griffiths group has infinite torsion.

Speaker: **Sergej Monavari** (Utrecht University)

Title: **Double nested Hilbert schemes of points and stable pair invariants**

Abstract: Hilbert schemes of points on a smooth projective curve are simply symmetric powers of the curve itself; they are smooth and we know almost everything about them. We generalize them by studying double nested Hilbert schemes of points, which parametrize flags of 0-dimensional subschemes satisfying certain nesting conditions dictated by Young diagrams. These moduli spaces are almost never smooth but admit a virtual fundamental class à la Behrend-Fantechi. We explain how this virtual structure plays a key role in computing stable pair invariants of local curves and (re)proving an instance of the conjectural correspondence with Gromov-Witten invariants.

Speaker: **Jean Fasel** (Institut Fourier, Grenoble)

Title: **How to (almost) construct low rank vector bundles on projective spaces**

Abstract: In this talk, we will discuss the problem to construct low rank
algebraic vector bundles on P^{n}. We will use motivic homotopy theory
to produce interesting classes of maps between projective spaces and
the classifying space BSL_{m} for n larger than m, and then try to
understand if these classes actually define vector bundles on these
projective spaces. Along the way, we will discuss algebraic spheres
and maps between them. This is joint work with A. Asok and M. Hopkins.

Speaker: **Zsolt Patakfalvi** (EPFL)

Title: **Replacing vanishing theorems in mixed characteristic and the Minimal Model Program for 3-folds over excellent schemes**

Abstract: Kodaira and Kawamata-Viehweg vanishing is frequently used to lift sections of adjoint bundles, a crucial part of many arguments in the classification theory of algebraic varieties, notably in many proofs of the Minimal Model Program. Although there is no counterexample known, we have strong indications that the above vanishing theorems fail in mixed characteristic situations, for example for 1.) proper, flat schemes over the p-adic numbers, or 2.) proper birational models of mixed characteristic local rings. I present a work that remedies this situation to some extent. In particular, we are able to a.) show Kodaira and Kawamata-Viehweg vanishing in many situations, b.) prove the 3-dimensional Minimal Model Program for excellent schemes, c.) draw geometric corollaries of point b.) to the existence of the moduli space of stable surfaces in mixed characteristic. This is a joint work with Bhargav Bhatt, Linquan Ma, Karl Schwede, Kevin Tucker, Joe Waldron and Jakub Witaszek.

Speaker: **Roberto Svaldi** (EPFL)

Title: **Minimal model program and foliations**

Abstract: A foliation on an algebraic variety is a partition of the variety into 'parallel' disjoint immersed complex submanifolds. Foliations naturally appears in a wide range of problems in algebraic geometry. I will explain recent progress in the birational classification of algebraic foliations in low dimension inspired by the theory of the Minimal Model Program. I will try to use key examples that exemplify the richness of the foliated world both in analogy and in opposition to the classical case of algebraic varieties. The talk will feature joint work with Calum Spicer.

Speaker: **Alessandro Chiodo** (Jussieu)

Title: **Spin structures on hyperelliptic graphs and the Z_{2}-quadratic form**

Abstract: We present briefly the theory of divisors on graphs, parallel to that of line bundles on curves. We show some tools for computing their ranks and we focus on the **Z**_{2}-quadratic forms defined by spin structures. We show how it generalizes in the case of hyperelliptic tropical spin curves. There, the quadratic form leads to a new picture. We show how it combines with the classical statement. This is work in collaboration with Marco Pacini.

Speaker: **Gavril Farkas** (Humboldt Universität Berlin)

Title: **Green's Conjecture via Koszul modules**

Abstract: Using ideas from geometric group theory we provide a novel approach to Green’s Conjecture on syzygies of canonical curves. Via a strong vanishing result for Koszul modules we deduce that a general canonical curve of genus g satisfies Green’s Conjecture when the characteristic is zero or at least (g+2)/2. Our results are new in positive characteristic (and answer positively the Eisenbud-Schreyer Conjecture), whereas in characteristic zero they provide a different proof for theorems first obtained by Voisin. Joint work with Aprodu, Papadima, Raicu and Weyman.