Oberseminar Algebraische Geometrie

News

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

Our next meeting is on Thursday, 18.01.2018 at 2:15 pm.

Schedule

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 forthcoming summer term 2018

Name Title Date
Christian Lehn tba 03.05.2018
Stefan Schreieder  tba 17.05.2018
Cinzia Casagrande tba 28.06.2018
Alessandra Sarti  tba 05.07.2018

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: Ulrich bundles on the intersection of two 4-dimensional quadrics

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: The Prym-Green conjecture for curves of odd genus

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: Brill-Noether theory of curves on abelian surfaces

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: Descent of numerically flat vector bundles and singular ball quotients

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.

 

   
© AG Schreyer