Seminar Algebra and Geometry - Spring 2022

It happens on Tuesdays from 10:30 to 12:00 in Spiegelgasse 1, Seminarraum 00.003 (ground floor). Some tea takes place before the talk, at 10:00 in Spiegelgasse 1, 6th floor.
March 1 Alexandra Kuznetsova
HSE/École polytechnique
Regularisations of pseudo-automorphisms I will give a condition for a positive entropy pseudo-automorphism of a threefold such that if this condition is satisfied the pseudo-automorphism can not be regularized. I will recall the construction of a positive entropy pseudo-automorphism introduced by J. Blanc and prove that for a very general choice of parameters this pseudo-automorphism satisfies the condition; thus, it is non-regularizable.
March 15 Quentin Posva
Gluing theory for slc varieties In the moduli theory of varieties, we are naturally led to consider a class of non-normal singularities, called semi-log canonical. This creates technical difficulties: for example, techniques from the MMP might fail on these non-normal varieties. Gluing theory was designed by Kollár to go back and forth between these singularities in characteristic zero and their normalization. I will report on my recent work on gluing theory in positive characteristic, and sketch some applications to birational geometry and moduli theory (e.g. abundance for surfaces, moduli of surfaces of general type in positive characteristic).
March 29 Zhixin Xie
Anticanonical geometry of the blow-up of ℙ4 in 8 points and its Fano model Mukai realised the blow-up X of ℙ4 in 8 points as a moduli space of vector bundles on a degree-one del Pezzo surface S. With the same construction, Casagrande-Codogni-Fanelli associated to S a smooth Fano fourfold Y with remarkable geometric properties, and described explicitly the interplay between S, X and Y. Building on their work, we continue to explore the birational geometry of Y which is an important example of Fano fourfold. We will describe completely the base scheme of the anticanonical system of Y, and discuss the action of the Bertini involution ιY on Y induced by the Bertini involution on S. In particular, we will explain the relation between ιY and the anticanonical map of Y, and show that ιY preserves every divisor in the anticanonical system of Y.
April 5 Mara Ungureanu
Counting points on projective curves For a curve in projective space, the varieties parametrising its secant planes are among the most studied objects in classical algebraic geometry. We shall review some of their basic properties and reformulate this extrinsic geometry problem in terms of objects intrinsic to the geometry of the abstract curve, namely secant divisors to a given linear series. We consider in particular enumerative formulas counting the number of points in the intersection of two such secant varieties on a given curve and discuss their validity. We shall see that the surprising behaviour of the counting formulas arises from the complicated geometry of secant varieties, which are not always of expected dimension.
April 12 Julie Déserti
The structure of the groups of germs of diffeomorphisms and their formal completions. In this talk I will give some properties of the group Gn of germs of diffeomorphisms of Cn. For instance if H is a subgroup of Gn, then H is residually finite, but also Hopfian. Furthermore, if n=1, then G1 is Hopfian, and the automorphism group of G1 can be described. This is a joint work with Dominique Cerveau.
April 26 Andrea Fanelli
A question on the real locus of smooth Fano threefolds Smooth complex Fano threefolds have been fully classified by Iskovskikh and Mori-Mukai and their description is well understood and rather explicit.
In this context it is natural to study the geometry of real smooth Fano threefolds and to investigate the connections between their algebraic properties and topological properties of their real loci.
In this talk I will survey some known results and present a joint work in progress with Frédéric Mangolte, in which we investigate the connectedness of the real locus for smooth Fano threefolds.
May 3 Gabriel Dill
Leibniz Universität Hannover
On morphisms between connected commutative algebraic groups Let K be a field of characteristic 0 and let G and H be connected commutative algebraic groups over K. We construct a natural retraction from the set of pointed morphisms of algebraic varieties from (G,0) to (H,0) to the set of homomorphisms of algebraic groups from G to H (for arbitrary G and H) which commutes with the composition and addition of morphisms, unique with this property. In particular, if G and H are isomorphic as algebraic varieties, then they are isomorphic as algebraic groups. This is false in positive characteristic or if K = ℂ and G and H are isomorphic only as complex-analytic varieties. If G has no non-trivial unipotent group as a direct factor, we give an explicit description of the sets of all morphisms and isomorphisms of algebraic varieties between G and H.
May 10 Dominique Mattei
Symplectic birational invollutions on hyperkähler manifolds of K3[n]-type. Hyperkähler manifolds (simply connected compact Kähler manifolds admitting a holomorphic symplectic form, HK for short) are higher dimensional analogues of K3 surfaces. These manifolds are interesting from several point of view (dynamical, arithmetical, geometric). In a joint work with Y. Dutta and Y. Prieto Montañez, we study finite order birational automorphism of HK manifolds deformation equivalent to the Hilbert scheme of points on a K3 surface. We prove that the existence of certain symplectic birational selfmaps implies that the HK manifold is birational to a moduli space of (twisted) sheaves on a K3 surface. Moreover, by means of stability conditions, we study the existence of symplectic birational involutions on such manifolds.
May 17 Pietro Beri
The Kodaira dimension of moduli spaces of irreducible holomorphic symplectic varieties Irreducible holomorphic symplectic (IHS) varieties are one of the three building blocks of varieties with trivial canonical bundle; they can be seen as a generalisation in higher dimension of K3 surfaces. In 2007, Gritsenko, Hulek and Sankaran proved that the moduli space of K3 surfaces of degree 2d is of general type when d>61: their strategy is to reduce the question to the existence of a certain cusp form for an orthogonal modular variety. In this talk, after a short introduction to IHS varieties, we give some motivations for the study of the Kodaira dimension of moduli spaces of IHS varieties, we sketch the reduction argument by Gritsenko Hulek and Sankaran and we explain how we used this method to prove general type results for some moduli spaces of IHS varieties in higher dimension. We also explain what the challenges are when we try to imitate their approach. Finally, we provide an infinite list of unirational moduli spaces (few of them are known). This is joint work in progress with I. Barros, E. Brakkee and L. Flapan.
May 24 Claudia Stadlmayr
Del Pezzo surfaces with global vector fields If X is a del Pezzo surface (or a weak del Pezzo, or an RDP del Pezzo), then its automorphism scheme Aut_X is a, possibly non-reduced, affine group scheme of finite type. In particular, X has infinitely many automorphisms if and only if Aut_X is positive-dimensional and then X admits global vector fields (since the space of global vector fields on X is the tangent space to the automorphism scheme). The last implication is an equivalence in characteristic 0, but its converse can fail in positive characteristic. Over the complex numbers, a del Pezzo surface with rational double point singularities admits global vector fields if and only if its minimal resolution, the corresponding weak del Pezzo surface, does. In small characteristics, one implication of this equivalence breaks down due to the existence of non-lifting vector fields on rational double points. I will explain how to overcome these obstacles in order to classify weak and RDP (if p \neq 2) del Pezzo surfaces with global vector fields. Further, I will show examples displaying interesting behaviour of such surfaces in small characteristics. This is joint work with Gebhard Martin.
May 31 Julia Schneider
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