MAGHI

More Algebraic Geometry at the Hausdorff Institute

This is a weekly research seminar in algebraic geometry organized by Laura Pertusi, Mirko Mauri, Raymond Cheng, and Sarah Frei as part of the Junior Trimester Program in Algebraic Geometry at the Hausdorff Research Institute for Mathematics. Meetings take place from 15:00 to 16:00, usually on Tuesdays, in the Hausdorff Institute Lecture Hall, Poppelsdorfer Allee 45.

26.09: Reinder Meinsma; Derived equivalence for elliptic K3 surfaces and Jacobians; We present a detailed study of Fourier-Mukai partners of elliptic K3 surfaces. One way to produce Fourier-Mukai partners of elliptic K3 surfaces is by taking Jacobians. We answer the question of whether every Fourier-Mukai partner is obtained in this way. This question was raised by Hassett and Tschinkel in 2015. We fully classify elliptic fibrations on Fourier-Mukai partners in terms of Hodge-theoretic data, similar to the Derived Torelli Theorem that describes Fourier-Mukai partners. This classification has an explicit computable form in Picard rank two, building on the work of Stellari and Van Geemen. We prove that for a large class of Picard rank 2 elliptic K3 surfaces all Fourier-Mukai partners are Jacobians. However, we also show that there exist many elliptic K3 surfaces with Fourier-Mukai partners which are not Jacobians of the original K3 surface. This is joint work with Evgeny Shinder.
10.10: Raymond Cheng; \(q\)-bic hypersurfaces; Let’s count: \(1\), \(2\), \(q+1\). The eponymous objects are special projective hypersurfaces of degree \(q+1\), where \(q\) is a power of the positive ground field characteristic. This talk will sketch an analogy between the geometry of \(q\)-bic hypersurfaces and that of quadric and cubic hypersurfaces. For instance, the moduli spaces of linear spaces in \(q\)-bics are smooth and themselves have rich geometry. In the case of \(q\)-bic threefolds, I will describe an analogue of result of Clemens and Griffiths, which relates the intermediate Jacobian of the \(q\)-bic with the Albanese of its surface of lines.
16.10: Tudor Ciurca; Irrationality of cubic threefolds in characteristic 2; In 1972 Clemens and Griffiths gave a formidable proof that a smooth cubic threefold over C is irrational. The proof was soon after adapted to any algebraically closed field of characteristic not 2 using algebraic methods. I will finish the story by extending the proof to the case of characteristic 2. As arithmetic applications, we answer a question of Deligne regarding arithmetic Torelli maps and establish the Shafarevich conjecture for cubic threefolds over function fields of characteristic 2.
20.10: Fei Xie; Quadric bundles over smooth surfaces; For a flat quadric bundle of relative even dimension with fibers of corank at most 1, there is a well established relation between its derived category and its relative Hilbert scheme of maximal isotropic subspaces (or its relative moduli of spinor bundles). For a smooth \(2m\)-fold with the structure of a quadric bundle over a smooth surface, there is a finite number of fibers with corank 2 and this relation fails. I will discuss how to fix the relation in this case.
23.10: Fumiaki Suzuki; Maximal linear spaces for pencils of quadrics and rationality; Over an arbitrary field \(k\) of odd characteristic, let \(X\) be a smooth complete intersection of two quadrics in \(\mathbf{P}^{2g+1}\). For every \(g\) at least 2, we show that the existence of a \((g-1)\)-plane, defined over \(k\), on \(X\) may be characterized by \(k\)-rationality of a certain 3-dimensional subvariety of the Fano scheme of \((g-2)\)-planes on \(X\), generalizing the \(g = 2\) case due to Hassett–Tschinkel and Benoist–Wittenberg. We also present a related result on \(k\)-rationality of the Fano schemes of non-maximal linear spaces on \(X\). This is joint work in progress with Lena Ji.
27.10: Shengxuan Liu; A note on spherical bundles on K3 surfaces; Let \(S\) be a K3 surface with the bounded derived category \(\mathrm{D}^b(S)\). Let \(E\) be a spherical object in \(\mathrm{D}^b(S)\). Then there always exists a non-zero object \(F\) satisfying \(\mathrm{RHom}(E,F)=0\). Further, there exists a spherical bundle \(E\) on some K3 surfaces that is unstable with respect to all polarization on \(S\). Also we “count” spherical bundles with a fixed Mukai vector. These provide (partial) answers to some questions of Huybrechts. This is a joint work with Chunyi Li.
31.10: Jack Petok; Zeta function of the K3 category of a cubic; We study the arithmetic of the K3 category associated to a cubic fourfold over a non-algebraically closed field \(k\). We start by constructing the Mukai structure of this K3 category with a natural action of Galois. For \(k\) a finite field, this lets us define the zeta function of a K3 category, an invariant under FM-equivalence of K3 categories. We provide a characterization of those cubic fourfolds whose K3 category has zeta function arising from a K3 surface defined over \(k\). One interesting outcome is that the zeta function does not always detect the geometricity of the K3 category. This is joint work with Asher Auel.
14.11: Stefano Filipazzi; On the boundedness of elliptic Calabi-Yau threefolds; In this talk, we will discuss the boundedness of Calabi-Yau threefolds admitting an elliptic fibration. First, we will review the notion of boundedness in birational geometry and its weak forms. Then, we will switch focus to Calabi-Yau varieties and discuss how the Kawamata-Morrison cone conjecture comes in the picture when studying boundedness properties for this class of varieties. To conclude, we will see how this circle of ideas applies to the case of elliptic Calabi-Yau threefolds. This talk is based on work joint with C.D. Hacon and R. Svaldi.
05.12: Mark de Cataldo; Geometry of moduli of t-connections; I will review some of my recent work on moduli spaces of Higgs bundles, of connections and of t-connections (which subsums both). These moduli spaces are objects of interest in the Non Abelian Hodge Theory of projective manifolds--warning: I will only discuss the case of curves--over the complex numbers and they have been intensively studied in the last thirty plus years. The situation over fields of positive characteristic is less explored and has also become the focus of what people call NAHT in characteristic p. While the moduli spaces of Higgs bundles and of connections are homeomorphic over the complex numbers, the situation over fields of positive characteristic is less clear. I will focus on explaining how a suitable compactification of these moduli spaces allows to bypass the lack of a homeomorphism to yield a canonical isomorphism of cohomology rings. Along the way, I will discuss some of the new phenomena, absent over the complex numbers, that emerge in positive characteristic. I will be short on technical details and my plan is to make the talk accessible to non-experts. For example, in illustrating the compactification technique, I will use as a guide the Ehresmann Lemma from differential topology. The talk contains some joint work with Siqing Zhang and with Davesh Maulik, Junliang Shen and Siqing Zhang.
12.12: Anna Abasheva; Surfaces with many algebraic structures; According to the GAGA principle, if two complex projective varieties are biholomorphic, then they are isomorphic as algebraic varieties. This is false for non-compact varieties; the first counterexample was due to Serre. Rodion Déev and I discovered a new example of a complex surface admitting countably many algebraic structures. Namely, we blow up the projective plane in sufficiently general nine points and remove the strict preimage of the elliptic curve passing through them. Our construction of the other algebraic structures relies on the Hopf transform -- 'surgery` that replaces a neighborhood of an elliptic curve with a neighborhood of a different elliptic curve.