# Algebraic geometry and moduli seminar

## Autumn Semester 2016

Note: The highlighted event marks the next occurring event and events marked with an asterisk (*) indicate that the time and/or location are different from the usual time and/or location.

Date / Time Speaker Title Location
16 September 2016
16:00-17:15
Prof. Dr. Ben Bakker
University of Georgia
HG G 43
Abstract: Classical Brill--Noether theory elegantly describes linear systems on generic curves; the problem of doing the same for sections of higher rank vector bundles has been extensively studied but the picture is far from complete. In joint work with G. Farkas, we prove a conjecture of Mercat for rank 2 bundles on generic curves classifying minimal slope bundles that admit a prescribed number of sections. As in the classical case, the Brill--Noether theoretic behavior of rank 2 bundles can be packaged into a "rank two Clifford index," but while the resulting stratification of M_g is different from the gonality stratification, we nonetheless show that the divisorial strata in both cases have slope 6+12/(g+1). The proof of the main theorem proceeds by specializing to curves on K3 surfaces, and time permitting we'll discuss the relationship with wall-crossing of Bridgeland stability conditions.
23 September 2016
16:00-17:15
Dr. Gergely Berczi
ETH Zürich
HG G 43
Abstract: I will start with a short report on recent progress in constructing quotients by actions of non-reductive algebraic groups and extending Mumford's geometric invariant theory to a wide class of non-reductive linear algebraic groups which we call graded groups. I will then explain how certain components of the Hilbert scheme of points on smooth varieties can be described as non-reductive quotients and why this description is especially efficient to study the topology of Hilbert schemes. In particular I will explain how equivariant localisation can be used to develop iterated residue formulae for tautological integrals on geometric subsets of Hilbert schemes and I present new closed formulae counting curves on surfaces (and more generally hypersurfaces in smooth varieties) with given singularity classes. This talk is based on joint work with Frances Kirwan and Andras Szenes.
28 September 2016
13:30-15:00
Dr. Alexander Buryak
ETH Zürich
HG G 43
Abstract: Using the intersection numbers of a given cohomological field theory with the double ramification cycles and the top Chern class of the Hodge bundle, one can construct a system of PDEs of certain type. We call this system the double ramification (DR) hierarchy. There is a natural way to associate a partition function to the DR hierarchy. Conjecturally, this partition function is related to the partition function of the cohomological field theory by a certain elementary transformation. Remarkably, the conjecture is equivalent to a system of relations between the correlators of the cohomological field theory that seems to be new. The talk is based on joint works with B. Dubrovin, J. Guere and P. Rossi.
30 September 2016
16:00-17:15
Dr. Hyenho Lho
ETH Zürich
HG G 43
Abstract: We calculate the genus two quasimap(stable quotient) invariants for local P^2 and verify the result predicted by physicists. It provides the first mathematical proof that quasimap(stable quotient) invariants are exactly same with the topological string amplitudes in B-model side for genus 2. If time is allowed, we discuss the way to understand holomorphic anomaly equations in terms of quasimap(stable quotient) invariants. This is joint work in progress with Rahul Pandharipande.
7 October 2016
16:00-17:15
Dr. Javier Fresan
ETH Zürich
HG G 43
Abstract: Exponential periods form a class of complex numbers containing the special values of the gamma and the Bessel functions, the Euler constant and other interesting numbers which are not expected to be periods in the usual sense of algebraic geometry. However, we can interpret them as coefficients of the comparison isomorphism between two cohomology theories associated to varieties with a potential: the de Rham cohomology of a connection with irregular singularities and the so-called “rapid decay” cohomology. I will explain how this point of view allows one to construct a Tannakian category of exponential motives and to produce Galois groups which conjecturally govern all algebraic relations among these numbers. No prior knowledge of motives will be assumed, and I will focus on examples rather than on the more abstracts aspects of the theory. This is a joint work with Peter Jossen.
12 October 2016
13:30-15:00
Junliang Shen
ETH Zürich
HG G 43
Abstract: By physical considerations, Huang, Katz and Klemm conjectured in 2015 that topological string partition functions for elliptic Calabi-Yau 3-folds are governed by certain Jacobi forms. This gives strong structure results for curve counting invariants of elliptic CY 3-folds. In the lecture series, I will explain a mathematical approach to prove (part of) the Huang-Katz-Klemm Conjecture. Our method is to use an involution on the derived category of the CY 3-fold X, and wall-crossing techniques developed by Toda. I will also discuss the connection to the Oberdieck-Pandharipande Conjecture, which concerns the enumerative geometry of the product of a K3 surface and an elliptic curve. For a fixed primitive curve class of genus h on K3, we prove that the generating series of curve counting invariants is a quasi-Jacobi form of index h-1 and weight -10. This is compatible with, and gives strong evidence for the OP Conjecture. The talks are based on joint work with Georg Oberdieck.
14 October 2016
16:00-17:15
Ignacio Barros
Humboldt Universität zu Berlin
HG G 43
Abstract: The Zariski closure of such spaces was recently understood and the task of giving a full birational classification is still in diapers. K3 surfaces come naturally into play when we want to construct rational curves through general points on this spaces. We use some Mukai models to prove unirationality of strata in small genus. The muse that serves us as role model is the space of odd theta characteristics, whose complete birational classification was successfully carried out by Farkas and Verra.
19 October 2016
13:30-15:00
Junliang Shen
ETH Zürich
HG G 43
Abstract: By physical considerations, Huang, Katz and Klemm conjectured in 2015 that topological string partition functions for elliptic Calabi-Yau 3-folds are governed by certain Jacobi forms. This gives strong structure results for curve counting invariants of elliptic CY 3-folds. In the lecture series, I will explain a mathematical approach to prove (part of) the Huang-Katz-Klemm Conjecture. Our method is to use an involution on the derived category of the CY 3-fold X, and wall-crossing techniques developed by Toda. I will also discuss the connection to the Oberdieck-Pandharipande Conjecture, which concerns the enumerative geometry of the product of a K3 surface and an elliptic curve. For a fixed primitive curve class of genus h on K3, we prove that the generating series of curve counting invariants is a quasi-Jacobi form of index h-1 and weight -10. This is compatible with, and gives strong evidence for the OP Conjecture. The talks are based on joint work with Georg Oberdieck.
21 October 2016
16:00-17:15
Johannes Schmitt
ETH Zürich
HG G 43
Abstract: We present the definition of the moduli spaces of twisted k-differentials, which are closed substacks of \bar M_g,n constructed by Farkas and Pandharipande. On the open part M_g,n they are defined by the condition that a weighted sum of the marked points agrees, as a divisor class, with the k-th power of the canonical line bundle of the curve. We give an indication how to compute the dimension of their components and a conjectural expression of their (weighted) fundamental class in terms of a tautological cycle studied by Pixton.
* 26 October 2016
14:00-15:00
Junliang Shen
ETH Zürich
HG G 43
Abstract: By physical considerations, Huang, Katz and Klemm conjectured in 2015 that topological string partition functions for elliptic Calabi-Yau 3-folds are governed by certain Jacobi forms. This gives strong structure results for curve counting invariants of elliptic CY 3-folds. In the lecture series, I will explain a mathematical approach to prove (part of) the Huang-Katz-Klemm Conjecture. Our method is to use an involution on the derived category of the CY 3-fold X, and wall-crossing techniques developed by Toda. I will also discuss the connection to the Oberdieck-Pandharipande Conjecture, which concerns the enumerative geometry of the product of a K3 surface and an elliptic curve. For a fixed primitive curve class of genus h on K3, we prove that the generating series of curve counting invariants is a quasi-Jacobi form of index h-1 and weight -10. This is compatible with, and gives strong evidence for the OP Conjecture. The talks are based on joint work with Georg Oberdieck.
28 October 2016
16:00-17:15
Dr. Jérémy Guéré
Humboldt Universität, Berlin
HG G 43
Abstract: In 2007, a quantum theory for quasi-homogeneous polynomial singularities was developed by Fan, Jarvis, and Ruan. It should be seen as the counterpart of Gromov-Witten (GW) theory for hypersurfaces in weighted projective spaces via the so-called Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence. In 2003, Lee established a K-theoretic version of GW theory. However, some aspects of GW theory, such as mirror symmetry, were still missing until last year. Indeed, Givental gave in 2015 a refined version called permutation equivariant GW K-theory, and proved some mirror symmetry statements in this new context, e.g. for the quintic hypersurface in P^4. In this talk, I will describe a joint work with Valentin Tonita and Yongbin Ruan, in which we define a K-theoretic version of the quantum singularity theory and we study its permutation equivariant part. I will focus on the quintic polynomial and explain how to prove mirror symmetry and the LG/CY correspondence.
2 November 2016
13:30-15:00
Junliang Shen
ETH Zürich
HG G 43
Abstract: By physical considerations, Huang, Katz and Klemm conjectured in 2015 that topological string partition functions for elliptic Calabi-Yau 3-folds are governed by certain Jacobi forms. This gives strong structure results for curve counting invariants of elliptic CY 3-folds. In the lecture series, I will explain a mathematical approach to prove (part of) the Huang-Katz-Klemm Conjecture. Our method is to use an involution on the derived category of the CY 3-fold X, and wall-crossing techniques developed by Toda. I will also discuss the connection to the Oberdieck-Pandharipande Conjecture, which concerns the enumerative geometry of the product of a K3 surface and an elliptic curve. For a fixed primitive curve class of genus h on K3, we prove that the generating series of curve counting invariants is a quasi-Jacobi form of index h-1 and weight -10. This is compatible with, and gives strong evidence for the OP Conjecture. The talks are based on joint work with Georg Oberdieck.
11 November 2016
16:00-17:15
Dr. Emanuel Scheidegger
Universität Freiburg
HG G 43
Abstract: Hypergeometric functions of order 3 and 4 appear as hemisphere partition functions of the A-model of the quartic K3 surface and the quintic threefold and as periods of the B-model of the mirror quartic and the mirror quintic, respectively. We discuss their analytic continuation to the singular or conifold point and its conjectured relation to the L-series of the quintic at the conifold point determined by Schoen given in terms of certain modular forms.
23 November 2016
13:30-15:00
Moscow State University
HG G 43
Abstract: For a germ of a quasihomogeneous function with an isolated critical point at the origin invariant with respect to an appropriate action of a finite abelian group (an admissible one), H.Fan, T.Jarvis, and Y.Ruan defined the so-called quantum cohomology group. This group is defined in terms of the vanishing cohomology groups of Milnor fibres of restrictions of the function to fixed point sets of elements of the group. The quantum cohomology group is considered as the main object of the so called quantum singularity theory or FJRW-theory. Fan, Jarvis, and Ruan studied some structures on the quantum cohomology group which generalize similar structures in the usual singularity theory. An important role in singularity theory is played by such concepts as the (integral) Milnor lattice, the monodromy operator, the Seifert form and the intersection form. Analogues of these concepts have not yet been considered in the FJRW-theory. We define an orbifold version of the monodromy operator on the quantum (co)homology group and a lattice which is invariant with respect to the orbifold monodromy operator and is considered as an orbifold version of the Milnor lattice. The action of the orbifold monodromy operator on it can be considered as an analogue of the integral monodromy operator. Moreover, we define orbifold versions of the Seifert form and of the intersection form. The talk is based on a joint work with W. Ebeling.
25 November 2016
16:00-17:15
Moscow State University
HG G 43
Abstract: One has a simple formula for the generating series of the Euler characteristics of the symmetric powers of a space: it is equal to the series $(1-t)^{-1}=1+t+t^2+\ldots$ not depending on the space in the exponent equal to the Euler characteristic of the space itself (I. Macdonald). A Macdonald type equation is a formula for the generating series of the values of an invariant for symmetric powers of spaces or varieties (or for their analogues) which gives this series as a series not depending on the space in the exponent equal to the value of the invariant for the space itself. If the invariant takes values in a ring different from the ring of integers, an expression of this sort requires an interpretation. It is given by the so called power structure over the ring. The most important example is a geometric power structure over the Grothendieck ring of complex quasiprojective varieties. One has a number of extensions of the concept of the Euler characteristic to the equivariant setting (i.e. for spaces with actions of a finite group) with values in the Burnside ring of the group and also to the generalized one with values in the Grothendieck ring of complex quasiprojective varieties. Also there exist the concept of orbifold Euler characteristic and its extensions. We shall discuss power structures over some (Grothendieck) rings and Macdonald type equations formulated in terms of them. The talk is based on joint works with I. Luengo and A. Melle-Hernandez.
7 December 2016
13:30-15:00
Dr. Ran Tessler
ETH Zürich
HG G 43
Abstract: I will describe a construction of genus 0 open GW and descendant theory for (P^1,RP^1), with an eye towards higher genus and other target pairs. If time permits the non stationary sector will be addressed. A joint work with Sasha Buryak, Rahul Pandharipande, and Amitai Zernik.
9 December 2016
16:00-17:15
Dr. Kaloyan Slavov
ETH Zürich
HG G 43
Abstract: Let X be an absolutely irreducible hypersurface of degree d in A^n, defined over a finite field F_q. The Lang--Weil bound gives an interval that contains #X(F_q). We exhibit an explicit interval, which does not contain #X(F_q), and which overlaps with the Lang--Weil interval for a certain range of q. The proof is elementary and uses a probabilistic counting argument. The talk will be accessible to a broad audience.
14 December 2016
14:00-15:30
Prof. Dr. John Ottem
Univ. of Oslo
HG G 43
Abstract: While the cones effective and nef divisors are fundamental objects in algebraic geometry, the corresponding cones for cycles of higher codimension have received much less attention and remain more mysterious. We survey some recent developments in this area, and in particular discuss the example of nef 2-cycles on the variety of lines on a cubic fourfold.
21 December 2016
13:30-15:00
Dr. Ran Tessler
ETH Zurich
HG G 43
Abstract: I will describe a construction of genus 0 open GW and descendant theory for (P^1,RP^1), with an eye towards higher genus and other target pairs. If time permits the non stationary sector will be addressed. A joint work with Sasha Buryak, Rahul Pandharipande, and Amitai Zernik.
13 January 2017
16:00-17:15
Prof. Dr. Bumsig Kim
KIAS (Seoul)
HG G 43
Abstract: We present a wall-crossing formula for the virtual classes of ε-stable quasimaps to GIT quotients and sketch its proof for complete intersections in projective space, with no positivity restrictions on their first Chern class. As a consequence, the wall-crossing formula relating the genus g descendant Gromov-Witten potential and the genus g epsilon-quasimap descendant potential is established. For the quintic threefold, our results may be interpreted as giving a rigorous and geometric interpretation of the holomorphic limit of the BCOV B-model partition function of the mirror family. This is a joint work with I. Ciocan-Fontanine.