Aspects of Moduli - Abstracts

 

Valery Alexeev:  Weighted stable hyperplane arrangements

We give a common generalization of (1) Hassett's weighted stable curves and (2) Hacking-Keel-Tevelev's stable hyperplane arrangements.

Jim Bryan:   Donaldson-Thomas and Gromov-Witten theory of Calabi-Yau orbifolds and resolutions

There are two basic theories of curve counting on Calabi-Yau threefolds. Donaldson-Thomas theory arises by considering curves as subschemes; Gromov-Witten theory arises by considering curves as the image of maps. Both theories can also be formulated for orbifolds. Let X be a dimension three Calabi-Yau orbifold and let Y-->X be a Calabi-Yau resolution. The Gromov-Witten theories of X and Y are related by the Crepant Resolution Conjecture. The Gromov-Witten and Donaldson-Thomas theories of Y are related by the famous MNOP conjecture. In this talk I will (with some

provisos) formulate the remaining equivalences: the crepant resolution conjecture in Donaldson-Thomas theory and the MNOP conjecture for orbifolds. I will discuss examples to illustrate and provide evidence for our

conjectures.

 

Paolo Cascini: On the Minimal Model Program

We will discuss some recent results and new aspects of the Minimal Model Program.

Lucia Caporaso:  Moduli of special line bundles on curves

I will describe some old and new results, together with some open questions, concerning the moduli space of "special" balanced line bundles on semistable curves.  As for smooth curves, a special line bundle is one whose space of global sections has dimension larger than expected.

Barbara Fantechi:  On Jun Li's proof of the degeneration formula

 

This is a report on joint work in progress with Dan Abramovich.  We take a new approach to Jun Li's result, based on the case of twisted structures along nodal singularities, with the aim of simplifying the proof and possibly extending the result to D-M stacks.

 

 

Gavril Farkas:  The birational geometry of moduli spaces of curves with level structure

Moduli of curves with level structure provide an interesting correspondence between the moduli spaces of curves and abelian varieties respectively.  Using Koszul-theoretic methods we prove that the moduli space R_g of Prym varieties of dimension g-1 is of general type for g>13 and that the moduli space S_g of spin curves is of general type for g>8. In contrast we show that S_g has negative Kodaira dimension for g<8.

Brendan Hassett:  Birational models of moduli spaces and geometric invariant theory

The Deligne-Mumford compactification of the moduli space of curves of genus g has many wonderful properties, but there are numerous alternatives that can be constructed using invariant theory and techniques from the minimal model program.  Examples include

        -pseudostable curves (due to D. Schubert)

        -weighted pointed stable curves of genus zero (with M. Simpson)

        -moduli of bicanonical curves (with D. Hyeon)

        -moduli of m-stable pointed curves of genus one (due to D. Smyth)

We focus on developing new modular interpretations of geometric invariant theory compactifications, which facilitate comparisons with existing spaces.  This yields new information on the birational structure (and especially the cone of ample divisors) of the Deligne-Mumford compactification.

 

Stefan Kebekus: An analogue of Shafarevich hyperbolicity for families over higher-dimensional base manifolds

Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type.

Given a quasi-projective threefold Y that maps to the moduli stack, we employ extension properties of logarithmic pluri-forms to establish a strong relationship between the moduli map and the minimal model program of Y: in all relevant cases the minimal model program leads to a fiber space whose fibration factors the moduli map. A much refined affirmative answer to Viehweg's conjecture for families over threefolds follows as a corollary. For families over surfaces, the moduli map can be often be described quite explicitly.

Sandor Kovacs:  Du Bois singularities and why you should care about them

In this talk I will recall what Du Bois singularities are and discuss their connections to better-known singularities, their importance for moduli problems, and recent advances related to them.

Radu Laza:  The moduli space of cubic fourfolds

 

Max Lieblich:  Moduli of orbifold sheaves and applications to arithmetic and non-commutative algebra

Stacks of stable sheaves on projective surfaces are known to have good regularity properties for large values of the discrete invariants: they are irreducible, normal, and lci.  I will discuss similar results for stacks of stable twisted sheaves on certain orbisurfaces, with an emphasis on their applications to 1) a basic problem about Brauer groups of function fields and 2) a classical question on the Hasse principle for geometrically rational varieties over global fields.

 

Marco Manetti:  Periods of generalized  deformations

A generalized deformation of a complex manifold is defined algebraically as a solution, up to gauge, of the Maurer-Cartan equation in the algebra of polyvector fields.  We show that in the Kaehler case,  every generalized deformation gives a canonical sequence of holomorphic maps into the Grassmannian of graded subspaces of the De Rham cohomology.  For classical deformations the above maps reduces to Griffiths period maps.

Kieran O'Grady:  Moduli and periods of double EPW-sextics

EPW-sextics are special sextic hypersurfaces in 5-dimensional projective space; such a sextic comes equipped with a

double cover - a double EPW-sextic. A generic double EPW-sextic is a smooth deformation of Hilb^2(K3). Double EPW-sextics are similar  to double planes branched over a sextic curve and also to varieties of lines  on cubic 4-folds. We will report on work in progress that aims to prove a Torelli Theorem for double EPW-sextics. We will also discuss relations with a conjectural birational Torelli Theorem for deformations of Hilb^2(K3).

 

 

Rahul Pandharipande:  Curve counting, derived categories, and the K3 surface

I'll talk about the enumerative geometry of curves on Calabi-Yau 3-folds X and the relationship to stable pairs in the derived category of X. The basic example for the lecture will be K3-fibered Calabi-Yau 3-folds. Connections will be made to the work of Kawai-Yoshioka on sheaves on K3 surfaces. I will explain our recent proof (with Klemm, Maulik, and Scheidegger) of the Yau-Zaslow conjecture for rational curve enumeration in all classes for K3 surfaces.

 

 

 

Yukinobu Toda:   Limit stable objects on Calabi-Yau 3-folds

The purpose of this talk is to introduce the notion of limit stability on the category of perverse coherent sheaves on Calabi-Yau 3-folds, and construct new enumerative invariants of curves via limit stable objects. I will show that such invariants are generalizations of stable pair invariants introduced by Pandharipande and Thomas. I will show that investigations of wall-crossing phenomena of limit stable objects lead to the solution of the rationality conjecture of stable pair invariants.

Gabriele Vezzosi:  Derived loop spaces and a categorical Chern character

I will describe the first steps of a theory of derived categorical sheaves in algebraic geometry, and show how derived loop spaces enable us to define a Chern character in this generalized setting.

Eckhart Viehweg (Universitat Duisburg-Essen):  Compactifications of smooth families and of moduli spaces of polarized manifolds

Click here  for Viehweg's abstract.

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