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Organizing Committee
Bucknell University
UFRO Universidad de La Frontera
Università Statale di Milano
Rijksuniversiteit Groningen
Université de Poitiers
Abstract
This introductory workshop offers a guided entry into the arithmetic, geometry and computation of K3 surfaces. Its aims are to establish a shared foundation for the semester, highlight the central role of K3 surfaces across several areas, and motivate deeper study. The workshop will introduce to the use of symbolic computation systems in algebraic geometry and in particular for computations on K3 surfaces such as giving equations for elliptic fibrations, rational curves, and automorphisms. The workshop will feature three minicourses -on geometry, arithmetic, and computation - together with general talks covering these interconnected topics.
The workshop and in particular the minicourses will be a perfect opportunity for PhD students and postdocs to get acquainted with different aspects and tools for the study of K3 surfaces.
The workshop and in particular the minicourses will be a perfect opportunity for PhD students and postdocs to get acquainted with different aspects and tools for the study of K3 surfaces.
Alessandra Sarti, Université de Poitiers
This opening workshop serves as an introduction for participants in the Fall Semester Program on "Computations on K3 Surfaces and Related Varieties." It is not open to outside applications.
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Workshop Schedule
Monday, September 14, 2026
8:30 - 8:50 AM
Check In
11th Floor Collaborative Space
8:50 - 9:00 AM
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:00 - 9:15 AM
Organizer Welcome
Opening Remarks - 11th Floor Lecture Hall
9:15 - 11:00 AM
Minicourse: K3 surfaces quotients of K3 surfaces
Tutorial - 11th Floor Lecture Hall
Alice Garbagnati, Università Statale di Milano
Abstract
The K3 surfaces are regular surfaces which admit an holomorphic symplectic form. We will consider finite order symplectic automorphisms: the minimal resolution of the quotient of a K3 surface by such an automorphism is another K3 surface.
This creates a relation between families (a priori different) of K3 surfaces, which can be described in terms of lattice polarized K3 surfaces. We will review the classical results for the symplectic involutions and we describe the more recent generalizations for the order 3 automorphisms.
Then, we will concentrate on specific subsets of K3 surfaces admitting symplectic involutions, characterized by the following property: the K3 surface which admits the symplectic automorphism and the one which is the desingularization of its quotient (by the symplectic automorphisms) lie in the same family of K3 surface. This property can be characterized in a lattice theoretically way, but it is not equivalent to be polarized with a prescribed lattice.
11:15 - 11:30 AM
Coffee Break
11th Floor Collaborative Space
11:30 AM - 12:30 PM
Minicourse: Density for rational points on special K3s
Tutorial - 11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
12:30 - 2:30 PM
Lunch/Free Time
2:30 - 3:15 PM
Cox rings of Calabi-Yau hypersurfaces in toric Fano varieties
11th Floor Lecture Hall
Michela Artebani, Universidad de Concepción
Abstract
This talk deals with Cox rings of Calabi-Yau varieties X which are general anticanonical hypersurfaces in smooth toric Fano varieties Z. We present two complementary results, formulated in terms of primitive pairs of the anticanonical polytope of Z. The first gives combinatorial conditions ensuring that X is a Mori dream space and provides an explicit presentation of its Cox ring. The second shows that certain relations among primitive pairs force Bir(X) to be infinite, hence X is not a Mori dream space.
As an application, we show that for Calabi-Yau hypersurfaces in dimensions two and three, either the Cox ring is finitely generated or the birational automorphism group is infinite. In the K3 case, where the Mori dream classification was already known via lattice theory, our approach gives a combinatorial interpretation together with explicit Cox ring presentations in the Mori dream cases.
This is joint work with Antonio Laface and Luca Ugaglia.
3:30 - 4:00 PM
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM
Open Collaboration Time
Open Collaboration Time
5:00 - 6:30 PM
Semester Program Welcome Reception
Reception - 11th Floor Collaborative Space
Tuesday, September 15, 2026
9:00 - 11:00 AM
Minicourse: Local/global principles, Brauer groups, obstructions, and Isogenies of K3 surfaces
Tutorial - 11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
11:15 - 11:30 AM
Coffee Break
11th Floor Collaborative Space
11:30 AM - 12:30 PM
Minicourse: Computing Picard Lattices of K3 Surfaces
Tutorial - 11th Floor Lecture Hall
Edgar Costa, MIT
Abstract
Computing the geometric Picard lattice of a K3 surface involves two complementary tasks: ruling out classes that cannot occur and
constructing enough divisor classes to generate the lattice. This minicourse will present approaches to both tasks.
The first part focuses on how the action of Frobenius constrains the specialization of the Picard lattice and provides upper bounds on the
geometric Picard rank. We will explore several approaches, including reductions modulo p and searching for p-adic obstructions to lifting divisor classes to characteristic zero.
The second part concerns ongoing work with Emre Can Sertöz focused on quartic K3 surfaces, which generates lower bounds on the geometric Picard rank. Starting from numerical period approximations, we identify putative divisor classes expected to be represented by smooth rational curves, and explain how to reconstruct and rigorously certify exact equations for the corresponding curves. These curves generate a saturated Galois-stable sublattice of the geometric Picard lattice; when its rank matches the upper bound obtained in the first part, we recover the full geometric Picard lattice as a Galois module.
12:30 - 2:30 PM
Lunch/Free Time
2:30 - 3:15 PM
Arithmetic Properties of K3 Surfaces with Large Automorphism Groups
11th Floor Lecture Hall
Joseph Silverman, Brown University
Abstract
I will discuss arithmetic properties of K3 surfaces of type (2,2,2) in P^1xP^1xP^1, with some remarks on the Markoff-Hurwitz surfaces x^2+y^2+z^2=axyz+k (which are affine log-K3 analogues) and higher dimensional Calabi-Yau varieties given by the vanishing of a (2,2,...,2) form in P^1xP^1x...xP^1.
3:30 - 4:00 PM
Coffee Break
11th Floor Collaborative Space
4:00 - 4:20 PM
The Prym-Clifford index
11th Floor Lecture Hall
Martina Miseri, Erdös Center (Alfréd Rényi Institute)
Abstract
Since the end of the 19th century Brill-Noether theory has studied how a curve can be mapped into a projective space. A classic and widely studied invariant for algebraic curves is the Clifford index, which encodes key aspects of the geometry of the canonical model of a curve. Based on a joint work with Margherita Lelli-Chiesa, I will introduce a new invariant for Prym curves, the Prym-Clifford index, that is, a Clifford index computed with respect to the tensor product of the canonical bundle and a non-trivial 2-torsion line bundle. I will present the main results of our project, which, in particular, show how this invariant captures the geometry of the curve. Depending on the time left, I will mention Nikulin surfaces, a special type of K3 surfaces, which are the natural candidates for finding examples of "Prym-exceptional" curves.
4:25 - 4:45 PM
Generalised Kummer surfaces of Jacobians of genus two curves
11th Floor Lecture Hall
Alvaro Gonzalez Hernandez, ICERM / Brown University
Abstract
Generalised Kummer surfaces are obtained by resolving the singularities of quotients of abelian surfaces, and are a useful source of examples of families of K3 surfaces with well-understood Néron-Severi lattices. In this talk, I will discuss the construction of explicit models for these quotients in the case where the abelian surfaces are Jacobians of genus two curves with interesting automorphism groups. My aim is to make the theory of generalised Kummer surfaces concrete and accessible, and to show that the arithmetic of these examples can be studied in a fairly hands-on way.
Wednesday, September 16, 2026
9:00 - 10:00 AM
Minicourse: K3 surfaces quotients of K3 surfaces
Tutorial - 11th Floor Lecture Hall
Alice Garbagnati, Università Statale di Milano
Abstract
The K3 surfaces are regular surfaces which admit an holomorphic symplectic form. We will consider finite order symplectic automorphisms: the minimal resolution of the quotient of a K3 surface by such an automorphism is another K3 surface.
This creates a relation between families (a priori different) of K3 surfaces, which can be described in terms of lattice polarized K3 surfaces. We will review the classical results for the symplectic involutions and we describe the more recent generalizations for the order 3 automorphisms.
Then, we will concentrate on specific subsets of K3 surfaces admitting symplectic involutions, characterized by the following property: the K3 surface which admits the symplectic automorphism and the one which is the desingularization of its quotient (by the symplectic automorphisms) lie in the same family of K3 surface. This property can be characterized in a lattice theoretically way, but it is not equivalent to be polarized with a prescribed lattice.
10:15 - 10:30 AM
Coffee Break
11th Floor Collaborative Space
10:30 AM - 12:30 PM
Minicourse: Computing Picard Lattices of K3 Surfaces
Tutorial - 11th Floor Lecture Hall
Edgar Costa, MIT
Abstract
Computing the geometric Picard lattice of a K3 surface involves two complementary tasks: ruling out classes that cannot occur and
constructing enough divisor classes to generate the lattice. This minicourse will present approaches to both tasks.
The first part focuses on how the action of Frobenius constrains the specialization of the Picard lattice and provides upper bounds on the
geometric Picard rank. We will explore several approaches, including reductions modulo p and searching for p-adic obstructions to lifting divisor classes to characteristic zero.
The second part concerns ongoing work with Emre Can Sertöz focused on quartic K3 surfaces, which generates lower bounds on the geometric Picard rank. Starting from numerical period approximations, we identify putative divisor classes expected to be represented by smooth rational curves, and explain how to reconstruct and rigorously certify exact equations for the corresponding curves. These curves generate a saturated Galois-stable sublattice of the geometric Picard lattice; when its rank matches the upper bound obtained in the first part, we recover the full geometric Picard lattice as a Galois module.
12:30 - 12:35 PM
Group Photo (Immediately After Talk)
11th Floor Lecture Hall
12:35 - 2:30 PM
Lunch/Free Time
2:30 - 3:15 PM
Minicourse Problem Session
Problem Session - 11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
3:30 - 4:00 PM
Coffee Break
11th Floor Collaborative Space
4:00 - 4:20 PM
Automorphisms of quartic surfaces and Cremona transformations
11th Floor Lecture Hall
Ana Victoria Martins Quedo, Università degli Studi di Ferrara
Abstract
Given a smooth quartic K3 surface $S\subset \mathbb{P}^3$, Gizatullin was interested {in which automorphisms of $S$ are induced by Cremona transformations of $\mathbb{P}^3$.} Later on, Oguiso answered it for some interesting examples, and he posed the following natural question:
Is every automorphism of finite order of any smooth quartic surface $S\subset \mathbb{P}^3$ induced by a Cremona transformation?
In this talk, we present joint work with Daniela Paiva, where we give a negative answer to the above question by constructing a family of smooth quartic K3 surfaces $S_n$ with Picard number two such that $Aut(S_n) = \mathbb{D}_{\infty}$ together with an involution of $S_n$ that is not derived by any element of $Bir(\mathbb{P}^3)$. More precisely, we will prove that no element of $Aut(S_n)$ is induced by an element of $Bir(\mathbb{P}^3)$.
4:25 - 4:45 PM
Transcendental Brauer Manin obstruction and prismatic cohomology
11th Floor Lecture Hall
Margherita Pagano, ICERM
Abstract
Recently, in a joint work with Emiliano Ambrosi and Rachel Newton, we gave an interpretation in terms of prismatic cohomology of Brauer classes that are relevant for the Brauer-Manin obstruction to the density of rational points in adelic points. In this talk I will give an overview on how this new perspective allow us to show that for most of the surfaces, having a non-zero action of Frobenius on de Rham cohomology in positive characteristic assures the existence of geometrically relevant Brauer classes in characteristic zero. I will focus on the consequences of this result on Enriques surfaces having good reduction at 2.
All is joint work with Emiliano Ambrosi
Thursday, September 17, 2026
9:00 - 10:00 AM
Minicourse Problem Session: K3 surfaces quotients of K3 surfaces
Problem Session - 11th Floor Lecture Hall
Alice Garbagnati, Università Statale di Milano
Abstract
The K3 surfaces are regular surfaces which admit an holomorphic symplectic form. We will consider finite order symplectic automorphisms: the minimal resolution of the quotient of a K3 surface by such an automorphism is another K3 surface.
This creates a relation between families (a priori different) of K3 surfaces, which can be described in terms of lattice polarized K3 surfaces. We will review the classical results for the symplectic involutions and we describe the more recent generalizations for the order 3 automorphisms.
Then, we will concentrate on specific subsets of K3 surfaces admitting symplectic involutions, characterized by the following property: the K3 surface which admits the symplectic automorphism and the one which is the desingularization of its quotient (by the symplectic automorphisms) lie in the same family of K3 surface. This property can be characterized in a lattice theoretically way, but it is not equivalent to be polarized with a prescribed lattice.
10:00 - 10:45 AM
Zeta functions of K3 surfaces over finite fields
11th Floor Lecture Hall
Kiran Kedlaya, University of California San Diego
Abstract
The zeta function of an algebraic variety over a finite field is a fundamental invariant which reflects many geometric properties. In the special case of K3 surfaces, we survey some of these connections; discuss techniques for computing zeta functions, both individually at scale; and indicate some open questions which we may see some progress on during the semester.
11:00 - 11:30 AM
Coffee Break
11th Floor Collaborative Space
11:30 AM - 12:15 PM
K3 surfaces that are triple covers of the plane
11th Floor Lecture Hall
Dino Festi, Università degli Studi di Napoli 'Federico II'
Abstract
It is well known that a K3 surface is a double cover of the projective plane if and only if it has degree 2; in this case, a projective model is indeed given by a double cover of the plane ramified above a smooth sextic curve.
It is then natural to move to the next step and ask when is a K3 surface a triple cover of the projective plane.
In this talk, we will show that there are six possible cases, classified by the genus of their generic hyperplane section.
We will provide a geometric construction of a projective model for three of them and show that one of them cannot be realized.
This is joint work in progress with Francesco Polizzi.
12:30 - 2:30 PM
Lunch/Free Time
2:30 - 3:15 PM
Mincourse Problem Session: Computing Picard Lattices of K3 Surfaces
Problem Session - 11th Floor Lecture Hall
Edgar Costa, MIT
Abstract
Computing the geometric Picard lattice of a K3 surface involves two complementary tasks: ruling out classes that cannot occur and
constructing enough divisor classes to generate the lattice. This minicourse will present approaches to both tasks.
The first part focuses on how the action of Frobenius constrains the specialization of the Picard lattice and provides upper bounds on the
geometric Picard rank. We will explore several approaches, including reductions modulo p and searching for p-adic obstructions to lifting divisor classes to characteristic zero.
The second part concerns ongoing work with Emre Can Sertöz focused on quartic K3 surfaces, which generates lower bounds on the geometric Picard rank. Starting from numerical period approximations, we identify putative divisor classes expected to be represented by smooth rational curves, and explain how to reconstruct and rigorously certify exact equations for the corresponding curves. These curves generate a saturated Galois-stable sublattice of the geometric Picard lattice; when its rank matches the upper bound obtained in the first part, we recover the full geometric Picard lattice as a Galois module.
3:30 - 4:00 PM
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM
Open Collaboration Time
Open Collaboration TIme
Friday, September 18, 2026
9:00 - 9:45 AM
Automorphisms of quartic surfaces and Cremona transformations II
11th Floor Lecture Hall
Carolina Araujo, IMPA
Abstract
In this talk, I will address the following question, attributed to Gizatullin: Which automorphisms of a smooth quartic surface in projective 3-space are restrictions of Cremona transformations of the ambient space? Techniques from birational geometry, and in particular the Minimal Model Program, provide powerful tools for studying this problem. I will report on recent progress on Gizatullin's problem, obtained in joint works with Alessio Corti and Alex Massarenti, and with Daniela Paiva and Sokratis Zikas.
10:00 - 10:45 AM
Logarithmic Enriques varieties
11th Floor Lecture Hall
Samuel Boissière, Université de Poitiers
Abstract
In a recent work in collaboration with Chiara Camere and Alessandra Sarti, we introduce logarithmic Enriques varieties as a singular analogue of Enriques manifolds, generalizing the notion of log-Enriques surfaces introduced by Zhang in 1991. In this talk, I will mainly focus on the possible torsion indices of the subfamily of logarithmic Enriques varieties that admit a quasi-étale cover by a singular symplectic variety, and I will give some examples.
11:00 - 11:30 AM
Coffee Break
11th Floor Collaborative Space
11:30 - 11:50 AM
How to find the most algebraic elliptic surfaces?
11th Floor Lecture Hall
Yilong Zhang, University of Georgia
Abstract
Vinberg’s two “most algebraic” K3 surfaces have Picard number 20 and transcendental lattices of discriminants 3 and 4, the two smallest possible values. What are their analogues among surfaces of Kodaira dimension one? In this talk, I will explore this question for a family of elliptic surfaces with p_g=q=1, whose second cohomology carries a Hodge structure of K3 type. This is joint work with François Greer.
11:50 AM - 12:10 PM
Automorphisms of transcendental value 60 on complex K3 surfaces
11th Floor Lecture Hall
Stevell Muller, Leibniz Universität Hannover
Abstract
In 1998, Machida and Oguiso proved that a finite order automorphism of a complex K3 surface cannot induce an order 60 action on the associated transcendental lattice. In 2024, Bayer-Fluckiger showed that there exist nonetheless projective K3 surfaces whose automorphism group has transcendental value 60. In this talk, we discuss the existence of infinite-order automorphisms of transcendental value 60 on K3 surfaces, and their dynamical degrees.
12:10 - 12:30 PM
Singular rational curves on Enriques surfaces
11th Floor Lecture Hall
Simone Pesatori, Erdős Center - HUN-REN Rényi Institute
Abstract
We show the first examples of integral rational curves of higher arithmetic genus on the general Enriques surface.
12:30 - 2:30 PM
Lunch/Free Time
2:30 - 3:15 PM
Density of degree d points on surfaces
11th Floor Lecture Hall
Isabel Vogt, Brown University
Abstract
Given a variety defined over a nonclosed field, the structure of the points defined over degree d extensions of the ground field reflects both the geometry and arithmetic of the variety. In this talk I will discuss some results and avenues for study about the Zariski density of such degree d points on surfaces, both building on our more complete picture for curves, and illustrating the difficulty in moving up in dimension. This talk is based on joint works with Viray, Chen--Church--Pasten, and Berg--Fu--Gazaki--Porzio--Rawson.
3:30 - 4:00 PM
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM
Open Collaboration Time
Open Collaboration Time
Application Information
This opening workshop serves as an introduction for participants in the Fall Semester Program on "Computations on K3 Surfaces and Related Varieties." It is not open to outside applications.
Applications are not currently open. Please check back at a later date.
Your Visit to ICERM
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