Schedule for: 18w5084 - Moduli Spaces: Birational Geometry and Wall Crossings

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Building on the work of Artin and Zhang, we will present a general construction of a proper moduli space parameterizing S-equivalences of semistable objects in an abelian category. Gieseker semistability and Bridgeland semistablity can both be viewed within this framework. This construction relies on a general theorem providing necessary and sufficient conditions for an algebraic stack to admit a good moduli space. This is joint work with Daniel Halpern-Leistner and Jochen Heinloth.

A del Pezzo fibration is one of the natural outputs of the Minimal Model Program for threefolds. At the same time, geometry of an arbitrary del Pezzo fibration can be unsatisfying due to the presence of non-integral fibers and singularities of an arbitrarily large index. In 1996, Corti developed a program of constructing `standard models' of del Pezzo fibrations within a fixed birational equivalence class. Standard models enjoy a variety of desired properties, one of which is that all of their fibers are $\mathbf Q$-Gorenstein integral del Pezzo surfaces. Corti proved the existence of standard models for del Pezzo fibrations of degree $d\geq 2$, with the case of $d=2$ being the most difficult. The case of $d=1$ remained a conjecture. In 1997, Kollár recast and improved the Corti’s result in degree $d=3$ using ideas from Geometric Invariant Theory for cubic surfaces. I will present a generalization of Kollár’s approach in which we develop notions of stability for families of low degree ($d\leq 2$) del Pezzo fibrations in terms of their syzygy points (i.e., relations among low degree equations cutting out del Pezzos). A correct choice of stability and a bit of enumerative geometry then leads to (very good) standard models in the sense of Corti. This is joint work in progress with Hamid Ahmadinezhad and Igor Krylov.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

The KSBA theory of stable log surfaces $(X,D)$ is the natural extension to dimension 2 of the Deligne-Mumford-Knudsen-Hassett moduli of pointed stable curves. In this talk I will describe a class of compactifications of the moduli space of elliptic surfaces constructed using as input both the KSBA theory as well as twisted stable maps to Deligne-Mumford stacks of Abramovich-Vistoli. These spaces exhibit wall-crossing phenomena as one varies the parameters of the moduli problem and are expected to interpolate between various previously studied compactifications. I will illustrate the case of rational elliptic surfaces and, if time permits, recent extensions to elliptic K3 surfaces. This talk is based on joint work with K. Ascher as well as work of G. Inchiostro.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Let $X$ be a complete intersection of three quadrics in projective space, of dimension at least three. We study these varieties and their moduli from the perspective of rationality, focusing on geometric and arithmetic structures that may induce rational parametrizations. (Joint with Pirutka and Tschinkel)

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Moduli spaces of sheaves on (or stable objects in the derived category of) higher-dimensional varieties are badly behaved in many ways. In the case of cubic threefolds and fourfolds (and other Fano varieties), one can get rid of these pathologies by restricting ourselves to stable objects that lie in a certain semiorthogonal component of the derived category, called Kuznetsov category. I will explain in examples how one can study concrete explicity geometry within this seemingly abstract setting. The underlying foundational results are the construction of stability conditions for the Kuznetsov category of cubic fourfolds (joint with Lahoz, Macri and Stellari), and the notion and construction of stability conditions in families of varieties (joint with Lahoz, Macri, Nuer, Stellari and Perry).

For fixed degree $d$, one could ask for a meaningful compactification of the moduli space of smooth degree $d$ surfaces in $\mathbb{P}^3$. Motivated by Hacking's work for plane curves, I will discuss a KSBA compactification of this space by considering a surface $S$ in $\mathbb{P}^3$ as a pair $(\mathbb{P}^3, S)$ satisfying certain properties. We will study an enlarged class of these pairs, including singular degenerations of both $S$ and the ambient space. The moduli space of the enlarged class of pairs will be the desired compactification and, as long as the degree $d$ is odd, we can give a rough classification of the objects on the boundary of the moduli space.

Moduli spaces related by a change of stability are often birational on a union of irreducible components and completely unrelated on unions of irreducible components. In order to compare virtual intersection numbers on such moduli spaces one needs to separate contributions from irreducible components. I will discuss a splitting of the virtual class of the moduli space of genus one stable maps. I hope that this technique is more general and will allow certain comparisons of virtual intersection numbers on moduli spaces obtained by a change of stability.

In this talk I will discuss basepoint free classes on the moduli space of stable pointed rational curves that arise as Chern classes of Verlinde bundles, constructed from integrable modules over affine Lie algebras, and the Gromov-Witten loci of smooth homogeneous varieties. We'll see that in the simplest cases these classes are equivalent. Examples and open problems will be discussed.

In this talk, I will describe several results on the cohomology of the general sheaf in a moduli space of sheaves on a projective surface. I will discuss joint work with Jack Huizenga on rational surfaces such as Hirzebruch surfaces and joint work with Howard Nuer and Kota Yoshioka on K3 surfaces.

Let X be a Calabi-Yau 3-dimensional orbifold, and let Y be a crepant resolution of the coarse moduli space of X. When X satisfies the "hard Lefschetz condition" (that is, when the fibres of the resolution Y are at most 1-dimensional), the Donaldson-Thomas crepant resolution conjecture of Bryan-Cadman-Young gives a precise relation between the DT curve counting generating functions of X and Y. I will explain a proof of this conjecture via the motivic Hall algebra and Joyce's wall crossing formula. This is joint work with Sjoerd Beentjes and John Calabrese

A cohomological field theory is a collection of cohomology classes on the moduli spaces of stable curves with marked points that are related by pullback along the natural maps between these moduli spaces. When these cohomology classes are taken with coefficients in a ring A, we can consider actions by derivations of the ring A and ask when these actions coincide with (an infinitesimal version of) Givental's R-matrix action. This provides a general framework for thinking about holomorphic anomaly equations, and I'll discuss some examples from the Gromov-Witten theory of an elliptic fibration. This is joint work with Georg Oberdieck.

I will review an axiomatic formulation of a 2D TQFT whose formalism is based on the edge-contraction operations on graphs drawn on a Riemann surface (cellular graphs). I will describe a new result, that ribbon graphs provide both cohomological field theory and a visual explanation of Frobenius-Hopf duality, that plays a crucial role in Givental-Teleman's classification theorem of CohFTs. No prerequisite is assumed. This is based on a work in progress with Motohico Mulase.

Mixed-spin-P-fields (MSP) theory integrates Gromov-Witten invariants and Fan-Jarvis-Ruan-Witten invariants of the Calabi-Yau quintic in a geometric way. Torus localisation on its virtual cycle defined via co-section localisation provides effective computations for higher genus GW invariants and FJRW invariants. Its variant NMSP is applied to prove BCOV’s Feynman Rule.

Understanding the structure of Gromov-Witten invariants of quintic threefolds is an important problem in enumerative geometry which has been studied since the early 90s. Together with Q. Chen and Y. Ruan, we are constructing new moduli spaces that we call "logarithmic GLSM moduli spaces". One application is toward proving conjectures from physics about higher genus Gromov-Witten invariants of quintic threefolds, such as the holomorphic anomaly equations. Another application, which also was the initial motivation to develop logarithmic GLSM, is toward proving a conjecture of myself, R. Pandharipande, A. Pixton and D. Zvonkine on loci of holomorphic differentials with prescribed zeros. In this talk, I will focus on the second application. Its main actor is the so-called Witten's r-spin class, the analog of the virtual class in the FJRW theory of the A_{r-1}-singularity.

Introduced by Schubert in the early nineties, the spaces of pseudo-stable curves provide modular compactifications for the spaces of smooth curves. In contrast to the usual Deligne-Mumford compactification by stable curves, pseudo-stability allows the curves to have both nodal and cuspidal singularities, but disallows elliptic tails. In this talk, I'll discuss recent work that aims to better understand the tautological intersection ring of the spaces of pseudo-stable curves.

A few years ago with Laza and Voisin we constructed a hyperkahler compactification of the intermediate Jacobian fibration associated to a general cubic fourfold. In this talk I will first show how a HK compactification J(X) exists for any smooth cubic fourfold X and then discuss how the birational geometry of the fibration is governed by any extra algebraic cohomology classes on X.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.