## Abstracts

### Fabrizio Catanese

#### Old and new constructions for surfaces of general type with p_g=q=2.

The classification of surfaces of general type with p_g=q=2 is an interesting and open chapter of surface theory.

For these, apart from the trivial cases where the Albanese image is a curve, respectively when K^2 attains its minimum value 4, we have examples with K^2=5,6,7,8 and with degree d of the Albanese map in the set {2,3,4,6}.

A component of their moduli space is said to be of the main stream if the map associating to a surface S its Albanese surface A= Alb(S) has image of dimension 3 (hence dominates a component of the moduli space of Abelian surfaces).

I will first show Penegini's examples of components not of the main stream, given by surfaces isogenous to a product (this is the only case where degree d=6 is attained).

I will then show very simple equations for some components of the main stream, named CHPP, PP4, AC3, due to the names of several authors, Chen-Hacon, Penegini-Polizzi, Alessandro, Catanese (here K^2=5,6 ).

Finally I shall describe how these components, in view of the Fourier Mukai transform, can be characterized via some assumption on the Albanese map.

### Stephen Coughlan

#### Lifting Artin Gorenstein rings

I report on joint work in progress with Grzegorz Kapustka and Michał Kapustka.

There is a conjectural classification of Artin Gorenstein rings in codimension four and regularity at most 5 (see talks of Hal Schenck and Beihui Yuan). We concentrate on the case of regularity 5 and study liftings of these Artinian rings to higher dimensional varieties.

Such liftings are curves with 3-divisible canonical class, or even surfaces, or canonical 3-folds or Calabi--Yau 4-folds etc...

The most difficult case is the maximal degree 30, and I will describe some of our results in this case.

### Tom Ducat

#### Quartic surfaces up to volume preserving equivalence

We consider log Calabi-Yau pairs of the form (PP^3, D), where D is a quartic surface, up to volume-preserving equivalence. The coregularity of the pair (PP^3, D) is a discrete volume-preserving invariant c=0,1 or 2, and which depends on the nature of the singularities of D. We classify all pairs (PP^3,D) of coregularity c=0 or 1 up to volume preserving equivalence. In particular, if c=0 then we show that (PP^3, D) admits a volume preserving birational map onto a toric pair.

### Sara Filippini

#### Residual intersections and Schubert varieties

The notion of residual intersections was introduced by Artin and Nagata. Roughly speaking, given an algebraic variety $X$ and a closed subscheme $Y$ in $X$, which is contained in another closed subscheme $Z$, then a closed subscheme $W$ such that $W \cup Y = Z$ is a residual intersection of $Y$ in $Z$.

This idea can be formalized as follows: Let $I$ be an ideal in a local Cohen-Macaulay ring $R$, and $A = (a_1, \ldots, a_s) \subsetneq I$. Then $J = A:I$ is called an s-residual intersection of $I$ if $ht(J) \geq s \geq ht(I)$. Residual intersections provide a generalization of linkage. Indeed, if $J = A:I$ and $I = A:J$ for $A$ a regular sequence, $I$ and $J$ are said to be linked.

I will show how results of Huneke and of Kustin and Ulrich on residual intersections for standard deteminantal ideals and Pfaffian ideals respectively arise in the context of ideals of Schubert varieties in the big opposite cell of homogeneous spaces. This is joint work with J. Torres and J. Weyman.

### Giulia Gugiatti

**Towards homological mirror symmetry for the Johnson-Kollár surfaces.**Homological mirror symmetry predicts a categorical equivalence between the complex geometry (the B-side) of a Fano variety and the symplectic geometry (the A-side) of its mirror.

In this talk I will discuss homological mirror symmetry for certain log del Pezzo surfaces, known as Johnson-Kollár surfaces, and their Hodge-theoretic mirrors. These surfaces fall out of the standard mirror constructions since they have empty anticanonical linear system. I will describe the derived category of coherent sheaves of the stacks associated to the surfaces, and I will discuss some preliminary results obtained on the A-side. The result on the B side is joint with Franco Rota, while the work on the A-side is in progress with Franco Rota and Matt Habermann.

### Angelo Felice Lopez

#### Varieties with Ulrich twisted normal or tangent bundles.

As is well known, it is a conjecture that any smooth variety X in P^N has an Ulrich vector bundle. Then a natural question arises: consider the bundles usually associated to X in P^N. When are they Ulrich up to twist?

Aside from trivial examples, it is easy to see that only N_X(-k) or N_X^*(k) or T_X(k) can be Ulrich for some integer k. For the first two, we will give an almost complete answer. As for the third one, the problem appears to be more difficult and we will present only some partial results. Work in collaboration with D. Raychadhury, A Casnati and V. Antonelli.

### Robert Pignatelli

#### Simple fibrations in (1,2)-surfaces and threefolds near the Noether line

This is a joint work with Stephen Coughlan, Yong Hu and Tong Zhang It has recently been proved that the volume of a threefold of general type is bounded from below from 4/3p_g-10/3 when p_g is big enough. This is known as the Noether inequality, 3-dimensional analog of the famous Noether inequality for the surfaces of general type. Surfaces/threefolds for which the equality holds are said "on the Noether line".

By classical works of Horikawa and others, we know that the canonical models of surfaces on the Noether line are exactly the fibrations over the projective line such that every fiber is "algebraically" like a smooth curve of genus 2. By analogy with this I and S. Coughlan introduced the concept of "simple" fibrations in (1,2) surfaces in math/2207.06845 and conjectured, also motivated by recent results of Y. Hu and T. Zhang, that all the threefolds "near" the Noether line are simple fibrations in (1,2) surfaces.

I will discuss our first results on this conjecture. As applications, I will provide an explicit description of these threefolds as hypersurfaces in a toric 4-fold, and a detailed description of the corresponding moduli spaces of canonical threefolds. I will also discuss the beautiful similarities with Horikawa's description of the moduli space of surfaces on the Noether line.

### Slawomir Rams

#### Low-degree rational curves on quasi-polarized K3-surfaces

I will present some upper bounds on the number of low degree rational curves on quasi-polarized K3 surfaces (joint with A. Degtyarev and M. Schuett)

### Miles Reid

#### Cayley cubics, Enriques sextics and applications

The audience should find most elements of my talk familiar and elementary:

(1) cubic surfaces of PP^3 with nodes at the 4 coordinate points

(2) sextic surfaces passing double through the 6 coordinate lines of PP^3

(3) the standard "monoidal" Cremona involution given by (x,y,z,t) |-> (1/x, 1/y, 1/z, 1/t)

(4) the Enriques-Fano 3-fold (PP^1 x PP^1 x PP^1)/(±1) with its eight 1/2(1,1,1) orbifold points

(5) toric constructions involving the face-centred cube

My main aim is to join these elements together into a single construction using graded ring methods and Mori theory.

### Eleonora Romano

#### Recent results on Fano varieties

In this talk we present some recent results on complex smooth Fano varieties. To this end, we first recall an invariant introduced by Casagrande, called Lefschetz defect. We review the literature to deduce that all Fano manifolds with Lefschetz defect greater than three are well known. Then we focus on the case in which the Lefschetz defect is equal to three, by discussing a structure theorem for such varieties. As an application, we use this result to classify all Fano 4-folds with Lefschetz defect equal to three: there are 18 families, among which 14 are toric. This is a joint work with C. Casagrande and S. Secci.

### Hal Schenck (online)

#### Gorenstein rings in codimension four and regularity at most five

I will discuss recent results and conjectures on Artin Gorenstein (AG) rings of codimension 4 and regularity at most 5. For regularity 3 and 4, we give a complete characterization of what betti tables are possible (there are 3 and 16, respectively), as well as relations in the parameter space (of the corresponding inverse quaternary cubic or quartic polynomial). While the Weak Lefschetz Property always holds for regularity at most 4, in regularity 5 it fails in interesting ways. We conjecture that for regularity 5 there are 36 possible Betti tables for AG rings, and show that for 4 of them the betti table type does not determine WLP. These results are based on the papers below. The talk by Beihui Yuan later in this conference will focus on the second two papers (which concern the regularity 4 case), and so most of my talk will focus on regularity 3 and regularity 5

arXiv https://arxiv.org/abs/

arXiv https://arxiv.org/abs/

arXiv https://arxiv.org/abs/

### Isabel Stenger

#### Cones of divisors on P^3 blown up at eight very general points

Let X be P^3 blown up at eight very general points. Then X is an example of a smooth projective threefold whose anticanonical divisor is nef but not semiample. In this talk we present an explicit description of the cone of nef divisors and the cone of effective divisors on X. Moreover, we show that a certain Weyl group acts with a rational polyhedral fundamental domain on the effective movable cone of X. This is a joint work with Z. Xie.

### Fabio Tanturri

**Zero loci of sections and Fano varieties**

If E is a vector bundle on an algebraic variety X, then for every global

section of E we can consider its zero locus. Zero loci are ubiquitous,

as many algebraic varieties can be realised as such. One interesting

feature is that many geometric properties of a zero locus can be deduced

from E and X, under some generality assumption. Starting from these

well-known facts, in this talk I will present a research programme based

on the explicit construction of families of interesting varieties as

zero loci of sections in a particular context. I will report on several

papers and ongoing projects which arose from this programme, mostly

concerning Fano varieties, shared with different people.

### Jerzy Weyman (online)

#### Gorenstein ideals of codimension 4.

I will discuss the structure theory of Gorenstein ideals of codimension 4 with n generators. It turns out that it is closely related to the root system of type E_n. In particular for n=7,8 I will discuss the examples related to Schubert varieties that should be generic. If time permits I will also discuss applications for general n.

### Jaroslaw Wisniewski

#### Geometric realization of birational maps and their resolutions via C* actions

The link between birational modifications and variation of quotients of C* actions is known for a few decades. I will discuss some new results regarding this concept. My talk is based on joint works with Gianluca Ochetta, Luis Sola Conde, and Eleonora Romano as well as Mateusz Michałek and Leonid Monin.

### Beihui Yuan

#### 16 Betti diagrams of Gorenstein Calabi-Yau varieties and a Betti stratification of Quaternary Quartic Forms

Motivated by the question of finding all possible projectively normal Calabi-Yau 3-folds in 7-dimensional projective spaces, we proved that there are 16 possible Betti diagrams for arithmetically Gorenstein ideals with regularity 4 and codimension 4. Among them, 8 Betti diagrams have been identified with those of Calabi-Yau 3-folds appeared in a list of 11 families founded by Coughlan-Golebiowski-Kapustka-Kapustka. Another 8 cannot be Betti diagrams of any smooth irreducible nondegenerate 3-fold. Based on the apolarity correspondence between Gorenstein ideals and homogeneous polynomials, and on our results on 16 Betti diagrams, we describe a stratification of the space of quartic forms in four variables.

This talk is based on the paper “Calabi-Yau threefolds in P^n and Gorenstein rings” by Hal Schenck, Mike Stillman and Beihui Yuan, and the preprint “Quaternary quartic forms and Gorenstein rings” by Michal Kapustka, Grzegorz Kapustka, Kristian Ranestad , Hal Schenck, Mike Stillman and Beihui Yuan.