Humboldt Universität zu Berlin
Mathem.-Naturwissenschaftliche Fakultät II
Institut für Mathematik

## Forschungsseminar "Algebraische Geometrie"

Wintersemester 2013/14

Das Forschungsseminar findet mittwochs in der Zeit von 15.00 - 17.00 Uhr in der Rudower Chaussee 25, 12489 Berlin-Adlershof, Raum 2.009 (Haus 2, Erdgeschoss), statt.

Seminar: Algebraic Geometry an der FU

 16.10.2013 Eduardo Esteves (IMPA, Rio de Janeiro) Title: Limit linear series Abstract: Linear series on a smooth, projective curve tell us about its projective geometry, whence their importance. Degeneration techniques abound in Algebraic Geometry. The term "limit linear series" was coined by Eisenbud and Harris in the 1980's; it describes degenerations of linear series along a family of curves degenerating to a (singular) curve of compact type. It arose naturally in the approach to proving the Brill-Noether Theorem. Several attempts have been made in the past 30 yearsto extend their theory to more general singular curves. In this talk I will describe my attempts at this, through several works by myself and in collaboration with Medeiros, Gatto, Pacini, Coelho, Salehyan, and lately with Osserman, Nigro and Rizzo. 23.10.2013 kein Seminar 30.10.2013 Vasudevan Srinivas (TIFR Tata Institute, Mumbai) Title: Etale Motivic cohomology and algebraic cycles Abstract: The Chow groups of smooth projective varieties may have torsion and co-torsion which are complicated, except in certain good cases like divisors or 0-cycles. In this talk I will discuss examples of these phenomena, and also variants, considered first by Lichtenbaum, which seem to have better properties in this respect. This is a report on some joint work with A. Rosenschon. 06.11.2013 kein Seminar (NoGAGS an der HU) 13.11.2013 Ananyo Dan (HU Berlin) Title: Geometry of the Noether-Lefschetz locus Abstract: One of the interesting questions in deformation theory is: When is a scheme parametrizing a family of smooth projective varieties non-reduced? One of the first examples in this direction was due to Mumford, where he gives an example of a non-reduced component of a Hilbert scheme parametrizing smooth projective space curves. In this talk I will discuss certain examples and criteria under which a scheme parametrizing a family of smooth projective curves or surfaces will be non-reduced. 20.11.2013 Arie Peterson (Universiteit van Amsterdam) Title: The moduli space of complex K3 surfaces Abstract: This talk will be about the moduli spaces of complex K3 surfaces. We are interested in their birational type (more narrowly, their Kodaira dimension) and their Picard group. Some strong results have been obtained in the past (most notably by Gritsenko, Hulek and Sankaran), using modular forms. We discuss some problems and partial solutions, in extending these results. 27.11.2013 John Christian Ottem (University of Cambridge) Title: Curves with positivity properties Abstract: A well-established principle in algebraic geometry is that geometric properties of an algebraic variety is rejected in the subvarieties which are in various senses 'positively embedded' in it. The primary example is the hyperplane section in a projective embedding of the variety, which gives rise to the notion of an ample divisor. However, in higher codimension it is less clear what it should mean in general for a subvariety to be 'positive'. We survey various definitions of positive embeddings and their geometric properties and discuss a related question of Peternell. 04.12.2013 Jarod Alper (Australian National University) Title: The Hassett-Keel program for Mg Abstract: The Hassett-Keel program for Mg is an attempt to provide modular descriptions for the log-canonical models of Mg by parameterizing certain classes of singular curves. In this talk, we will outline an intrinsic approach to construct the modular interpretations of the first contraction and the first two flips extending the work of Hassett and Hyeon. The emphasis in this talk will be on discussing various heuristics that allow one to predict which curves to parameterize. This is joint work with Maksym Fedorchuk, David Smyth and Fred van der Wyck. 11.12.2013 Alessandro Verra (Università Roma Tre) Title: On the universal principally polarized abelian variety of dimension 5 and on the slope of A6 Abstract: Let Ag be the moduli space of principally polarized abelian varieties of dimension g. In the talk it is proven that the universal ppav over A5 is a unirational variety U that dominates the boundary of the perfect cone compactification of A5. Using some covering families of rational curves in U the slope of A6 is studied and a bound on this slope is obtained. Joint work with Gavril Farkas. 18.12.2013 kein Seminar 08.01.2014 Talks by applicants for the IRTG postdoc-position 15:00 - 16:30 Zavosh Amir-Khosravi (Toronto) Title: An Isomorphism of Moduli Stacks Induced by Serre’s Tensor Construction Abstract: Let K ⊂ C be a quadratic imaginary number ﬁeld, n > 0 an integer, and Mn the moduli K space of abelian schemes A over S, for S an OK -scheme, which are equipped with an action of OK and an OK -linear principal polarization, such that A has relative dimension n over S, and the action of OK on LieS ( A) coincides with the action induced by the structure map of S over OK . If Hermn denotes the category of ﬁnite projective OK -hermitian modules of rank n which are positive-deﬁnite and non-degenerate, there is an isomorphism of stacks of the form ∼ Hermn ⊗Herm1 M1 − Mn K → K induced by Serre’s tensor construction. We will explain this statement, and if time permits describe how the ideas involved may be more generally used to construct cycles on moduli spaces that are arithmetic models of PEL Shimura varieties. 16:30 - 18:00 Rachel Newton (MPI) Title: Computing transcendental Brauer groups of products of CM elliptic curves Abstract: In 1971, Manin showed that the Brauer group Br(X) of a variety X over a number field K can obstruct the Hasse principle on X. In other words, the existence of points everywhere locally on X despite the lack of a global point is sometimes explained by non-trivial elements in Br(X). Since Manin's observation, the Brauer group has been the subject of a great deal of research. The 'algebraic' part of the Brauer group is the part which becomes trivial upon base change to an algebraic closure of K. It is generally easier to handle than the remaining 'transcendental' part and a substantial portion of the literature is devoted to its study. In contrast, until recently the transcendental part of the Brauer group had not been computed for a single variety. The transcendental part of the Brauer group is known to have arithmetic importance – it can give non-trivial obstructions to the Hasse principle and weak approximation. I will use class field theory together with results of Ieronymou, Skorobogatov and Zarhin to compute the transcendental part of the Brauer group of the product ExE for an elliptic curve E with complex multiplication. The results for the odd-order torsion descend to the Kummer surface Kum(ExE). MONTAG 13.01.2014 Robert Lazarsfeld (Stony Brook University) Beginn 13:00 Uhr, Raum 3.006 (Rudower Chaussee 25) Title: Asymptotic syzygies of curves and higher-dimensional varieties Abstract: I will discuss some joint work with Lawrence Ein concerning the asymptotic behavior of the syzygies associated to curves and higher-dimensional varieties as the positivity of the embedding line bundle grows. In the case of curves, I will discuss how a variant of the Hilbert scheme techniques introduced by Voisin leads (among other things) to a surprisingly quick proof of the conjecture that one can read off the gonality of a curve from the syzygies of any one embedding of sufficiently large degree. Time permitting, I will also survey some results suggesting that in higher dimensions, the syzygies that occur are "as complicated as possible." 15.01.2014 Rahul Pandharipande (ETH Zürich) Title: Curves in imprimitive classes on K3 surfaces and the Katz-Klemm-Vafa formula Abstract: I will explain our recent proof (with R. Thomas) of the KKV formula governing higher genus curve counting in arbitrary classes on K3 surfaces. The subject intertwines Gromov-Witten, Noether-Lefschetz and Donaldson-Thomas theories. A tour of these ideas will be included in the talk. 22.01.2014 Talks by applicants for the IRTG postdoc-position 15:00 - 16:30 Benjamin Bakker (Courant) Title: The geometric Frey-Mazur conjecture Abstract: A crucial step in the proof of Fermat's last theorem was Frey's insight that a nontrivial solution would yield an elliptic curve with modular p-torsion but which was itself not modular. The connection between an elliptic curve and its p-torsion is very deep: a conjecture of Frey and Mazur, stating that the p-torsion group scheme actually determines the elliptic curve up to isogeny (at least when p>13), implies an asymptotic generalization of Fermat's last theorem. We study geometric analogs of this conjecture, and show that over function fields the map from isogeny classes of elliptic curves to their p-torsion group scheme is one-to-one. Our proof involves understanding curves on certain Shimura varieties, and fundamentally uses the interaction between its hyperbolic and algebraic properties. This is joint work with Jacob Tsimerman. 29.01.2014 Tamás Szamuely (Renyi Institute Budapest) Title: Principal homogeneous spaces over p-adic curves Abstract: Let C be a smooth projective curve defined over a field k. When k is algebraically closed, a classical result of Steinberg implies that every C-family of principal homogeneous spaces under a reductive group has a section over some open subset of C. In recent joint work with David Harari, we are interested in the case where k is a p-adic field. Here such sections do not exist in general, even if we impose the existence of local analytic sections around each point. However, we show that in a large number of cases the absence of global rational sections can be explained by a cohomological obstruction. 05.02.2014 kein Seminar 12.02.2014 Slawomir Rams (Jagiellonian University Krakow und Leibniz Universität Hannover) Title: On quartics with many lines Abstract: We discuss some aspects of geometry of quartic surfaces with lines of the second kind. We sketch the proof of the fact that a smooth quartic surface over an algebraically closed field of different from 2, 3 contains at most 64 lines (joint work with Matthias Schuett). FREITAG 07.03.2014 Loring Tu (Tufts University und MPI Bonn) Beginn 15:15 Uhr, Raum 2.009 Title: Computing integrals using fixed points Abstract: Many invariants in algebraic geometry, differential geometry, and topology can be represented as integrals. For example, according to the Gauss--Bonnet theorem, the Euler characteristic of a compact oriented surface in $\R^3$ is $1/2\pi$ times the integral of the curvature form. In general, integrals are notoriously difficult to compute. However, when there is a group acting on a compact oriented manifold, the equivariant localization formula of Atiyah--Bott--Berline--Vergne converts the integral of an equivariantly closed form into a finite sum over the fixed points of the action, thus providing a powerful computational tool. An integral can also be viewed as a pushforward map from a manifold to a point, and in this guise it is intimately related to the Gysin homomorphism. This talk will highlight two applications of the equivariant localization formula. We will show how to use it to compute characteristic numbers of a homogeneous space and to derive a formula for the Gysin map of a fiber bundle.

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Sommersemester 2013