Mukai (1988) classified maximal finite groups acting symplectically on a K3 surface and showed that the largest one is the Mathieu group \(M_{20}\), of order 960. Then Kondo (1999), Bonnafé-Sarti (2021) and Brandhorst-Hashimoto (2021) complemented this study showing the possible extensions of \(M_{20}\) acting on a K3 surface and giving explicit examples of these surfaces. In this talk we will discuss about K3 surfaces admitting the action of \(M_{20}\), in order to prove that there exist infinitely many, and we will relate about the same problem for IHS manifolds which are Hilbert schemes of 2 points on a K3 surface. This is a joint work with Romain Demelle (Université de Poitiers) and Pablo Quezada Mora (Universidad de La Frontera).

A smooth degree \(d\) del Pezzo surface is obtained by blowing up the projective plane at \((9-d)\) generic points. Their tropicalizations are polyhedral complexes recording valuation spaces on these surfaces. These objects can be used to construct nice compactifications of the algebraic surfaces and their moduli. In this talk, we will discuss how to tropicalize these surfaces (and their moduli) and their families of lines as we vary the input points we blow up, and the combinatorial computational challenges that arise when doing so. This is based on joint works with Anand Deopurkar, Amanda Knetch and Cecilia Salgado.

The Cone Conjecture of Morrison-Kawamata (already proved in the case of surfaces) states that the Nef cone of any surface S such that \(K_S=0\) has a polyhedral fundamental domain, by considering the action by automorfisms of S. The same result is not true for pairs (S,D) such that D is a divisor and \(K_S+D=0\). For instance, take S the blowup of \(P^2\) in 9 very general points and D the cubic through the 9 points. In this case, \(Aut(S)\) is trivial, while the Nef cone is not polyhedral. By the other hand, we will see that it is possible to construct a polyhedral fundamental domain for the Nef cone of S, if we consider an other action – the so called Cremona action. The same happens for a half of the Nef cone if we blowup more points. In this talk, we will explain this construction and discuss possible generalizations for the Cone Conjecture of Morrison-Kawamata for pairs.

T-singularities are the two-dimensional cyclic quotient singularities that admit a Q-Gorensetein smoothing. These are precisely the singularities showing up in a normal degeneration of canonical surfaces in the KSBA moduli space. If we fix \(K^2\) and χ , then there is a finite list of the possible T-singularities in the surfaces parametrized by the KSBA moduli space. Explicitly writing down this list is a complicated question. Previously Rana and Urzúa have classified the case of surfaces with one T-singularity. In this talk, we will discuss the case of surfaces with multiple T-singularities. We will give general bounds and how it can be improved in special cases. This talk is based on joint work with J. Rana and G. Urzúa.

We discuss the Mori dream space (MDS) property for blow-ups of toric surfaces defined by rational plane triangles at a general point. We consider a parameter space of all such triangles and show how it can be used to prove the MDS property for these varieties. Furthermore, this parameter space helps explain most known results in the area and has also led to surprising new results, including examples of such surfaces with a semi-open effective cone. This is joint work with José Luis González and Kalle Karu.

Given a smooth quartic surface \(S\subset \mathbb{P}^3\), Gizatullin was interested in which automorphisms of \(S\) are induced by Cremona transformations of \(\mathbb{P}^3\). As a consequence of a Takahashi’s result, we know that if the surface \(S\) satisfies that any curve of degree \(<16\) is a complete intersection of \(S\) with another hypersurface of \(\mathbb{P}^3\), then every non-trivial automorphism of \(S\) are not induced by \(Bir(\mathbb{P}^3)\setminus Aut(\mathbb{P}^3)\). In this talk, we will focus in the case where the smooth quartic surface \(S\subset \mathbb{P}^3\) has Picard number two, and it contains curves of degree \(<16\) which are not complete intersections. Also, we will give examples of such surfaces where every automorphism of \(S\) is induced by Cremona transformations.

The problem of characterizing the Cox rings of a given variety has been thoroughly studied in the last years, for example, in the case of a K3 Mori dream surface of rank 2 (Ottem, 2013), of a K3 Mori dream surface of rank 3 (Artebani, Correa Deisler, Laface, 2021), and of a rational jacobian elliptic surface (Artebani, Garbagnati, Laface, 2016). A possible generalization in that last case is debilitating the hypothesis on jacobianity. On this talk, we will discuss how the techniques of Artebani, Correa Deisler and Laface can be adapted to this case and how this allows us to give a characterization of the degrees that appear in the Cox rings of these surfaces, and also to specialize it to the case of Halphen index 2. This is a joint work with Michela Artebani.

In this talk, we will discuss the variation of the Mordell-Weil rank on fibers of elliptic surfaces. We say that there is a rank jump within fibers above closed points of the base when their Mordell-Weil rank is strictly larger than the generic rank. Inspired by Néron’s constructions, we will present ideas on how to use and choose multisections for the fibration and use them to show that the rank jumps infinitely often. This is a specialized version of my plenary talk, where I will present the proofs of the main results presented there.

The moduli space of marked cubic surfaces is one of the most classical moduli spaces in algebraic geometry, dating back to the nineteenth century work of Cayley and Salmon studying the 27 lines on a cubic surface. Compactifications of this moduli space have been intensively studied from numerous perspectives, including tropical geometry, Hodge theory, Geometric Invariant Theory, classical algebraic geometry, K-moduli theory, and KSBA moduli theory. I will describe the interplay between these various compactifications via a detailed description of the \(W(E_6)\)-invariant birational geometry of the moduli space of marked cubic surfaces. This description reveals several new \(W(E_6)\)-equivariant compactifications, whose potential modular interpretations are as-yet undiscovered. Furthermore, I will discuss various aspects of the geometry of these compactifications including an explicit description of the degenerate surfaces that appear, and an explicit presentation of the Chow rings of all of these compactifications.

Let \(X\) be a smooth surface, we denote by \(Pic(X)\) \(\otimes\) \(\mathbb{R}\) the tensor product of the Picard group of \(X\) and \(\mathbb{R}\). Let \(Nef(X)\) \(\subseteq\) \(Eff(X)\) \(\subseteq\) \(Pic(X)\) \(\otimes\) \(\mathbb{R}\) be the nef and effective cone of \(X\) respectively. We consider the Mori cone of \(X\) as the closure of \(Eff(X)\).

Suppose that the class of the canonical divisor of \(X\), \(K\), is nonzero. The Cone Theorem establishes that subspace \(K_{<0}\) of the Mori cone of \(X\), i.e., all elements of \(\overline{Eff}(X)\) such that their intersection product with \(K\) is negative, are generated by a contable number of \((-1)-\)curves. So far, there are no results describing the rest of the Mori cone for an arbitrary smooth surface.

A special case is when \(X\) is the blow-up of \(\mathbb{P}^2\) on a set of \(s>0\) points in very general position, which we denote by \(X_s\). The Mori cone of \(X_s\) has been widely studied because it is related to several famous conjectures such as the Nagata conjecture [7], the conjecture of \((-1)\)-curves [5, Conjecture 1.1] and the equivalent conjectures of Segre, Harbourne, Gimigliano and Hirschowitz (SHGH conjecture) [4]. If \(s \leq 9\), \(\overline{Eff}(X_s)\) can be described as the sum of certain rays, while if \(s \geq 10\), the \(K\)-nonnegative part remains unknown.

In [1], C. Ciliberto, R. Miranda and J. Roé determine certain extremal rays of the Mori cone of \(X_s\) in \(K^{\perp}\) for all \(s\geq 10\) and in \(K_{>0}\) for all \(s\geq 13\). In [2], the authors study \(Nef(X_s)\) in more detail and show the existence of \(8\)-dimensional rational spheres at the boundary \(\overline{Eff}(X_s)\) for all \(s\geq 13\). The existence of these rays proves that the Mori and nef cones of \(X_s\) are not rationally polyhedral, and, moreover, gives evidence for the Strong \(\Delta\)-Conjecture [3, Conjecture 3.2.9] which implies Nagata’s conjecture.

In the thesis supervised by Professor Antonio Laface and in collaboration with Professor Luca Ugaglia, we have determined new reformulations of the \((-1)\)-curves and SHGH conjectures, in addition to obtaining some results similar to those stated in [2] motivated by the study of \(Nef(X_{10})\).

References:

[1] C. Ciliberto, R. Miranda, and J. Roé, Irrational nef rays at the boundary of the Mori cone for very general blowups of the plane, arXiv e-prints (2022), arXiv:2201.08634.

[2] , Quadric cones on the boundary of the Mori cone for very general blowups of the plane, 2023.

[3] C. Ciliberto, B. Harbourne, R. Miranda, and J. Roé, Variations on Nagata’s Conjecture, arXiv: Algebraic Geometry (2012).

[4] C. Ciliberto and R. Miranda, The Segre and Harbourne-Hirschowitz conjectures, Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001), NATO Sci. Ser. II Math. Phys. Chem., vol. 36, Kluwer Acad. Publ., Dordrecht, 2001, pp. 37–51. MR1866894

[5] T. de Fernex, On the Mori cone of blow-ups of the plane, arXiv: 2010arXiv1001.5243D (2010).

[6] R. Lazarsfeld, Positivity in algebraic geometry I : classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete 48, Springer, 2004.

[7] M. Nagata, On the 14-th Problem of Hilbert, American Journal of Mathematics 81 (1959), 766.