Singular Log Structures and Fano Threefolds

• Alessio Corti

Room: 02.04. Kjølv Egelands hus KE E-264; 03.04. Kjell Arholms hus KA U-136; 04.04. Kjølv Egelands hus KE A-202 We introduce a class of singular log schemes in 3-dimensions and conjecture that log schemes in this class admit log crepant log resolutions. We provide some examples as evidence and relate the conjecture to the conjecture made joint work with Filip&Petracci, the Gross-Siebert program and the construction of Fano 3-folds.

Fano moduli for curves and quivers

• Pieter Belmans

Room: 02.04. Kjølv Egelands hus KE E-264; 03.04. Kjell Arholms hus KA U-136; 04.04. Kjølv Egelands hus KE A-202 Moduli spaces of vector bundles on a curve are a classical and well-studied class of Fano varieties, with many interesting properties. Moduli spaces of quiver representations of an acyclic quiver are likewise (almost) Fano, with many interesting properties. There exists a rich dictionary between the two types of moduli spaces, explaining how phenomena for moduli of bundles have an analogue for moduli of representations, and vice versa. Sometimes the analogy informs us what to expect on the curve side, sometimes it informs us what to expect on the quiver side. I will survey aspects of this dictionary, and describe some important open problems for these.

Tom & Jerry triples unprojection format

• Vasiliki Petrotou

Room: 02.04. Kjølv Egelands hus KE E-264; 03.04. Kjell Arholms hus KA U-136; 04.04. Kjølv Egelands hus KE A-202 Unprojection is a theory introduced by Miles Reid which constructs more complicated Gorenstein rings starting from simpler data. This theory has found many applications in algebraic geometry and also in algebraic combinatorics. Tom and Jerry are two formats of unprojection defined and named by Miles Reid which lead to the construction of codimension 4 Gorenstein rings starting from codimension 3 Gorenstein rings. In this talk, we introduce a new unprojection format which we call Tom & Jerry triples. This format, generalizes the Tom and Jerry unprojection formats and constructs codimension 6 Gorenstein rings starting from codimension 3 Gorenstein rings. As an application, we will construct two families of codimension 6 Fano 3-folds in weighted projective space which appear in the Graded Ring Database.

Categorical Torelli theorems for weighted hypersurfaces and the Veronese double cone

• Jørgen Rennemo

Room: 02.04. Kjølv Egelands hus KE E-264; 03.04. Kjell Arholms hus KA U-136; 04.04. Kjølv Egelands hus KE A-202 The derived category of a Fano variety will often have a semiorthogonal decomposition consisting of a sequence of exceptional line bundles and a remaining subcategory, which we will call the Kuznetsov category. The categorical Torelli problem asks whether knowing this category up to exact equivalence is enough to determine the variety up to isomorphism. We will explain how a technique based on a computation of the Hochschild-Serre algebra of the category shows that the answer is yes in many cases, including the case of a Veronese double cone. This is joint work with Xun Lin and Shizhuo Zhang.

Fano 3-Fold weighted Hypersurfaces

• Jihun Park

Room: 02.04. Kjølv Egelands hus KE E-264; 03.04. Kjell Arholms hus KA U-136; 04.04. Kjølv Egelands hus KE A-202 In this talk, we will explore the 130 families of Fano 3-fold weighted hypersurfaces, with a particular focus on their non-rationality and K-stability. This is an ongoing collaboration with T. Okada.

K-stability of Fano weighted hypersurfaces

• Taro Sano

Room: 02.04. Kjølv Egelands hus KE E-264; 03.04. Kjell Arholms hus KA U-136; 04.04. Kjølv Egelands hus KE A-202 K-stability (or existence of Kähler-Einstein metrics) of explicit Fano varieties has been studied for a long time. Delta invariants (stability thresholds) detect the K-stability of Fano varieties. Moreover, Abban--Zhuang developed a powerful method to compute the delta invariants by adjunctions. In this talk, I will explain our recent results on the K-stability of some Fano weighted hypersurfaces via the Abban--Zhuang method.

New birational invariants for Fanos

• Ludmil Katzarkov

Room: 02.04. Kjølv Egelands hus KE E-264; 03.04. Kjell Arholms hus KA U-136; 04.04. Kjølv Egelands hus KE A-202 In this talk we will introduce new birational invariants inspired by HMS and CFT. Many examples of nonrational and G nonrational invariants will be introduced.

Cluster type variety

• Joaquin Moraga

Room: 02.04. Kjølv Egelands hus KE E-264; 03.04. Kjell Arholms hus KA U-136; 04.04. Kjølv Egelands hus KE A-202 Toric varieties are ubiquitous objects in algebraic geometry. A toric variety can be described as a (partial) compactification of an algebraic torus such that the volume form of the torus has poles along every divisor added in the compactification. In this talk, we will introduce the concept of cluster type varieties which are (partial) compactifications of algebraic tori such that the volume form has at most poles and no zeros along the boundary. These varieties have been considered by people in Mirror Symmetry, especially by Alessio Corti. In this talk, we will discuss questions regarding the detection of these varieties and their appearance in the classification of Fano varieties.

Birational mirrors to pairs of Laurent polynomials

• Hannah Tillmann-Morris

Room: 02.04. Kjølv Egelands hus KE E-264; 03.04. Kjell Arholms hus KA U-136; 04.04. Kjølv Egelands hus KE A-202 Under mirror symmetry, deformation classes of Fano varieties are associated to mutation classes of maximally mutable Laurent polynomials (MMLPs). We expect birational relationships between the general members of two deformation classes to be reflected, in the pair of mirror mutation classes, as combinatorial relationships between two MMLPs. I will present an alternative construction of a Fano variety that is mirror to a given MMLP, which uses mirror theorems of the Gross-Siebert program. The resulting Fano variety is difficult to describe explicitly. However, when two given MMLPs are related by certain combinatorial conditions, the construction can be extended to include the construction of a birational map between the two Fano varieties produced. Applying all this to rigid MMLPs in two variables recovers all but one of the blow-ups in the chain of smooth del Pezzo surfaces.

Some log singular surfaces with unobstructed deformations

• Simon Felten

Room: 02.04. Kjølv Egelands hus KE E-264; 03.04. Kjell Arholms hus KA U-136; 04.04. Kjølv Egelands hus KE A-202 The logarithmic Bogomolov-Tian-Todorov theorem guarantees that infinitesimal deformations of a proper log Calabi--Yau space over a log point are unobstructed once certain cohomological conditions are met. We know that these conditions are satisfied for log smooth and more generally log toroidal spaces. In this talk, I report ongoing joint work with Matthias Zach where we check the cohomological conditions for some families with more general log singularities by means of explicit computation.

Enumerative geometry of quantum periods

• Tim Gräfnitz

Room: 02.04. Kjølv Egelands hus KE E-264; 03.04. Kjell Arholms hus KA U-136; 04.04. Kjølv Egelands hus KE A-202 I talk about joint work with Helge Ruddat, Eric Zaslow and Benjamin Zhou interpreting the q-refined theta function of a log Calabi-Yau surface as a natural q-refinement of the open mirror map, defined by quantum periods of mirror curves for outer Aganagic-Vafa branes on the local Calabi-Yau threefold. The series coefficients are all-genus logarithmic two-point invariants, directly extending the relation found by the first three authors. The main part of the proof is combinatorial in nature, using a convolution relation for Bell polynomials, and thus works in any dimension. We find an explicit discrepancy at higher genus in the relation to open Gromov-Witten invariants of the Aganagic-Vafa brane, expressible in terms of relative invariants of an elliptic curve.

Relative mirror symmetry and some applications

• Fenglong You

Room: 02.04. Kjølv Egelands hus KE E-264; 03.04. Kjell Arholms hus KA U-136; 04.04. Kjølv Egelands hus KE A-202 We consider mirror symmetry for a log Calabi--Yau pair (X,D), where X is a Fano variety and D is an anticanonical divisor of X. The mirror of the pair (X,D) is a Landau--Ginzburg model (X^\vee, W), where W is a function called the superpotential. Following the mirror constructions in the Gross--Sibert program, W can be described in terms of Gromov--Witten invariants of (X,D). I will explain a relative mirror theorem for the pair (X,D). As applications, I will explain how to compute the superpotential W, the classical period of W and more.

LSD Fanos

• Michel van Garrel

Room: 02.04. Kjølv Egelands hus KE E-264; 03.04. Kjell Arholms hus KA U-136; 04.04. Kjølv Egelands hus KE A-202 Condition LSD (log smooth degeneration) states that the pair (Y,D) of Y=Fano and D=smooth anticanonical divisor admits a normal crossings degeneration to a union of blow ups of toric varieties. Condition HDTV states that Y may be constructed via the HDTV mirror construction, a special case of intrinsic mirror symmetry. The expectation is that LSD <=> HDTV. Moreover, they imply the existence of an interesting function, shown to be modular in some cases, and special properties of Apéry numbers of Y. Even, there are some hints of links to rationality questions. Beyond dimension 2, a lot of this remains conjectural at best. Nonetheless, given the confluence of experts, this is an ideal place to discuss some preliminary results and paths forward.

Lefschetz hyperplane theorem for weighted projective spaces and cohomological invariants of Fano hypersurfaces

• Anna-Maria Raukh

Room: 02.04. Kjølv Egelands hus KE E-264; 03.04. Kjell Arholms hus KA U-136; 04.04. Kjølv Egelands hus KE A-202 Given a generic hypersurface of a fixed degree in a n-dimensional weighted projective space, we describe the cohomology groups H^k for k<n and establish an explicit formula for the pullback map. Our results extend classical Lefschetz hyperplane theorem to weighted projective spaces and cohomology with integer coefficients. These results can be applied to compute cohomological invariants of Fano 3-folds of this form.

Laurent polynomials and deformations of non-isolated Gorenstein toric singularities

• Matej Filip

Room: 02.04. Kjølv Egelands hus KE E-264; 03.04. Kjell Arholms hus KA U-136; 04.04. Kjølv Egelands hus KE A-202 We establish a correspondence between one-parameter deformations of an affine Gorenstein toric variety X, defined by a polytope P, and mutations of a Laurent polynomial f, whose Newton polytope is equal to P. If the Newton polytope P of f is two dimensional and there exists a set of mutations of f that mutate P to a smooth polygon, then, we show that the Gorenstein toric variety, defined by P, admits a smoothing. This smoothing is obtained by proving that the corresponding one-parameter deformation families are unobstructed and that the general fiber of this deformation family is smooth.