## Oberseminar Algebraische Geometrie (AG Lazić / AG Schreyer)

News

We meet regularly thursdays in Saarbrücken, E2 4, SR 10 starting at 4:00 pm.

Schedule

 Name Title Date Zsolt Patakfalvi Positivity of the Chow-Mumford line bundle for families of K-stable Q-Fano varieties 26.04.2018 Christian Lehn tba 03.05.2018 No seminar Christi Himmelfahrt 10.05.2018 Stefan Schreieder tba 17.05.2018 Christian Liedtke A Néron-Ogg-Shafarevich criterion for K3 Surfaces 24.05.2018 No seminar Fronleichnam 31.05.2018 Hanieh Keneshlou Unirational component of the moduli stack of 6-gonal genus 11 curves 07.06.2018 14.06.2018 Hanieh Keneshlou The unirationality of the Hurwitz spaces H10,8 and H13,7 21.06.2018 Cinzia Casagrande tba 28.06.2018 Alessandra Sarti tba 05.07.2018

Zsolt Patakfalvi: The Chow-Mumford (CM) line bundle is a functorial line bundle on the base of any family of polarized varieties, in particular on the base of families of Q-Fano varieties (that is, Fano varieties with klt singularities). It is conjectured that it yields a polarization on the conjectured moduli space of K-semi-stable Q-Fano varieties. This boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with  K-semi-stable/K-polystable Q-Fano fibers. I present a joint work with Giulio Codogni where we prove the necessary semi-positivity statements in the K-semi-stable situation, and the necessary positivity statements in the uniform K-stable situation, including in both cases variants  assuming K-stability  only for very general fibers. Our statements work in the most general singular situation (klt singularities), and the proofs are algebraic, except the computation of the limit of a sequence of real numbers via the central limit theorem of probability theory. I also  present a birational geometry application to the classification of Fano varieties.

Christian Liedtke: Given a family of smooth and projective manifolds over a pointed disk, one may ask whether this family has good reduction, that is, whether this family can be extended to a smooth family over the whole disk. A necessary condition for this is that the monodromy actions on cohomology, that is, the actions of the fundamental group of the pointed disk on the singular cohomology groups of a general fiber, is trivial. But does the converse hold? First, we generalise the setup as follows: let R be a complete local ring with field of fractions K and residue field k (Spec K generalises the pointed disk and Spec R generalises the whole disk), and let X be a smooth and projective variety over K. In this setup, good reduction translates into finding a smooth and proper scheme over Spec R with generic fiber X. Moreover, trivial monodromy translates into the action of the absolute Galois group of K on all l-adic cohomology groups H^*(\bar{X},Q_l) being unramified (or crystalline if l=char(k)).
For Abelian varieties, it is classical known by a theorem of Serre and Tate (and already established by Néron, Ogg, and Shafaravich for elliptic curves) that good reduction is equivalent to having unramified/crystalline Galois-actions on H^1.
In my talk, I will first introduce the above notions and then, I will talk on joint work with Matsumoto, Chiarellotto, and Lazda, where we study K3 surfaces over K and ask whether good reduction is equivalent to having unramfied/crystalline Galois-actions on H^2. It turns out that this is almost true, but that the right analog of a Néron-Ogg-Shafarevich criterion for K3 surfaces is rather subtle.

Schedule of the winter term 2017/18

 Name Title Date Pagona Koulakidou Divisors as fibres of morphisms 19.10.17 Nikolaos Tsakanikas On varieties birational to abelian varieties George H. Hitching Tangent cones to generalised theta divisors and generic injectivity of the theta map 07.12.17 Gianluca Pacienza Density of Noether-Lefschetz loci of irreducible holomorphic symplectic varieties and applications 18.01.18 Andrea Fanelli Del Pezzo fibrations in positive characteristic 25.01.18 Luca Tasin A non-vanishing result for weighted complete intersections 01.02.18

Abstracts

Luca Tasin: Let X be a smooth (or mildly singular) projective variety and let H be an ample line bundle on X. Kawamata conjectured that if H-K_X is ample, then the linear system |H| is not empty. I will explain that the conjecture holds true for weighted complete intersections which are Fano or Calabi-Yau, relating it with the Frobenius coin problem. This is based on a joint work with M. Pizzato and T. Sano.

Andrea Fanelli: In this talk, I will discuss some pathologies for the generic fibre of del Pezzo fibrations in characteristic p>0, motivated by the recent developments of the MMP in positive characteristic. The main application of the joint work with Stefan Schröer concerns 3-dimensional Mori fibre spaces.

Gianluca Pacienza: We will try to illustrate how useful such density results can be by presenting several (old and new) applications to: the existence of rational curves on projective IHS varieties, the study of relevant cones of divisors, the study of lagrangian fibrations and a refinement of Hassett's result on cubic fourfolds whose Fano variety of lines is isomorphic to to an Hilbert scheme of 2 points on a K3 surface. We also discuss Voisin's conjecture on the existence of coisotropic subvarieties on IHS varieties and relate it to a stronger statement on Noether-Lefschetz loci in their moduli spaces.

George H. Hitching: Let C be a Petri general curve of genus g. The tangent cones of the Riemann theta divisor on Pic^{g-1}(C) have been used in various ways by Kempf and Schreyer and by Ciliberto and Sernesi to give new proofs of Torelli's theorem. We use a related approach to study the generalised theta divisor D(V) of a semistable bundle V over C of rank r and integral slope. For large enough g, we show how a sufficiently general such V can be reconstructed from the tangent cone of D(V) at a suitable singular point. We use this to give a constructive proof and a sharpening of Brivio and Verra's theorem that the theta map from the moduli space of semistable bundles of rank r and trivial determinant to the projective space |r\Theta| is generically injective for large values of g. (Joint work with Michael Hoff)

Schedule of the summer term 2017

 Name Title Date Christian Bopp Moduli of lattice polarized K3 surfaces via relative canonical resolutions 18.05. Sascha Blug Die Picard Gruppe von hyperelliptischen Kurven 01.06. Jonas Baltes Der Riemannsche Abbildungssatz und Metriken konstanter Krümmung auf Riemannschen Flächen Janik Schug Puiseux-Reihen 08.06. Christian Ikenmeyer Formula size, iterated matrix multiplication, and algebraic geometry Andreas L. Knutsen Brill-Noether theory of curves on abelian surfaces 29.06. Thomas Peternell Descent of numerically flat vector bundles and singular ball quotients Hanieh Keneshlou On Accola's genus bound for algebraic curves 06.07. Frank-Olaf Schreyer Horrocks splitting on products of projective spaces Michael Kemeny The Prym-Green conjecture for curves of odd genus 03.08. Yeongrak Kim Ulrich bundles on the intersection of two 4-dimensional quadrics 09.08. Jonas Baltes Der Riemannsche Abbildungssatz und Metriken konstanter Krümmung auf Riemannschen Flächen (Teil II) 24.08. Janik Schug Puiseuxreihen und ihre Anwendungen 05.10.

Yeongrak Kim: A coherent sheaf F on a projective variety X is Ulrich if its pushforward by a finite degree map is trivial. Since they naturally appears in several different theories, the study of Ulrich bundles becomes important. In this talk, I will discuss two different approaches to construct Ulrich bundles on the intersection of two 4-dimensional quadrics: via Serre correspondence and via derived categories. I will also briefly explain a connection between generalized theta series. This is a joint work with Y. Cho and K.-S.Lee.

Michael Kemeny: We will present a proof of the Prym-Green conjecture on the resolution of a paracanonical curve of odd genus and arbitrary torsion level. The proof proceeds by using curves on ruled surfaces over an elliptic curve. These surfaces naturally arise as desingularizations of limiting K3 surfaces with elliptic singularities, and come up in Arbarello-Bruno-Sernesi's study of the Wahl map and deformations of the cone. They have the downside of being irregular, which makes the study of syzygies more complicated than for K3s, but on the upside they allow for inductive arguments on the genus of the curve, which is not possible for a K3. Joint with Gabi Farkas.

Andreas L. Knutsen: The Brill-Noether theory of curves on K3 surfaces is well understood. Until recently, quite little has been known for curves on abelian
surfaces. In the talk I will present some recent results obtained with M. Lelli-Chiesa and G. Mongardi.

In particular, we show that the general curve in the linear system |L| on a general primitively polarized abelian surface (S,L) is Brill-Noether
general, as in the K3 case. However, contrary to the K3 case, there are  smooth curves in |L| possessing "unexpected" linear series, that is, with negative Brill-Noether number. As an application, we obtain the existence of components of special Brill-Noether loci of the expected dimension in the moduli space of curves.

Thomas Peternell: In my talk I will explain recent results with Greb, Kebekus and Taji concerning the uniformization of klt spaces whose (orbifold) Chern classes are extremal in the sense that they satisfy the Miyaoka-Yau equality.