Seminars
Abstract: We talk about "unlikely intersections" whenever we have a non-empty
intersection between algebraic varieties that, for dimensional reasons, we do not expect to
intersect. This expectation lies behind several landmark results and conjectures in Diophantine
geometry, including Faltings’ Theorem (formerly the Mordell Conjecture), the Manin-Mumford
Conjecture, the André-Oort Conjecture (proved by Pila, Shankar and Tsimerman), and the still open
Zilber-Pink Conjecture. In this talk, I will give an introduction to unlikely intersections,
focusing first on algebraic tori and abelian varieties, and then on families of abelian varieties.
I will survey some key results by Masser-Zannier and Barroero-Capuano in this setting, and finally
present results from my PhD thesis establishing some partial progress on the Zilber-Pink
Conjecture for curves in abelian schemes.
Abstract: We present the construction of distinguished non-Kähler metrics on
non-compact Calabi-Yau 3-folds. These metrics solve a system of equations known as the IIB system
which arises in theoretical physics and is related to recent attempts to define notions of
"canonical" metrics on non-Kähler Calabi-Yau manifolds. The examples we construct include
infinitely many complete metrics obtained by deforming an asymptotically conical Calabi-Yau
3-folds in the direction of a non-trivial Äppli class and families of solutions on the ordinary
double point and its smoothing that enjoy a cohomogeneity one symmetry (i.e. there is a symmetry
group that acts with 1-dimensional orbit space) and produce a "conifold transition" in non-Kähler
Calabi-Yau geometry. The talk is based on joint work with Mario Garcia-Fernandez.
Abstract: In 1972, Serre proved that the Galois representations arising from the
p-power torsion points of non-CM elliptic curves over \(\mathbb{Q}\) have open image in
\(\operatorname{GL}_2(\mathbb{Z}_p)\), and Mazur later initiated a vast programme to determine all
such possible images explicitly -- for fixed p, it is known that there are only finitely many
possibilities. Much progress has been made for small primes p, but a complete classification
remains open beyond \(p \in \{2,3,13,17\}\).
In this talk, I will describe recent progress on this problem for p = 7, based on a surprising
correspondence between rational points on modular curves and primitive integer solutions to
certain generalised Fermat equations of signature (2,3,7), such as \( a^2 + 28b^3 = 27c^7. \) We
show that these Diophantine equations can be reduced to determining the rational points on a
finite collection of genus-3 curves. As a consequence, we are able for example to determine the
rational points on a modular curve of genus 69 and establish that the 7-adic Galois images of
elliptic curves over \(\mathbb{Q}\) are determined by their reduction modulo \(7^2\).
Abstract: In join work with Manon Parent we have investigated how small the
\(L^2\)-norm of an exponential polynomial can get if we fix the norm of the its coefficient
vector. We prove lower and upper bounds of this minimum in terms of the degree of the polynomial
and we apply our methods to a variant of a problem of Hilbert. We prove that on any given interval
the infimum of the \(L^2\)-norm of exponential polynomials with integer coefficients on this
interval is 0.
Abstract: In 1919, while studying birational automorphisms of the projective
plane, A. Coble introduced a new family of surfaces, today known as Coble surfaces. We start by
showing some basic facts, and underling the properties which link these rational surfaces to the
world of Enriques surfaces. Next, we present some recent results on automorphisms of a Coble
surface, with a particular attention to the case of involutions. We show that, under suitable
generic assumptions, any involution on a Coble surface is a lift of a Bertini involution.
The content of this talk is based on my PhD thesis, written under the supervision of Prof. A.
Verra.
Abstract: A variety is retract rational if the solutions of its defining equation
can be parametrized by rational functions. Examples are provided by rational or stably rational
varieties. We explain some history on the problem of deciding whether a given variety is (retract)
rational, concentrating mostly on the case of cubics. We will then explain how recent joint work
with Engel and de Gaay Fortman on the failure of the integral Hodge conjecture for abelian
varieties implies by earlier work of Voisin that very general cubic threefolds over the field of
complex numbers are not retract rational, hence not stably rational.
Abstract: We explore the intersection of the Hassett divisor C8, parametrizing
smooth cubic fourfolds X containing a plane P with other divisors \(C_i\). Notably we study the
irreducible components of the intersections with \(C_{12}\) and \(C_{20}\). These two divisors
generically parametrize respectively cubics containing a smooth cubic scroll, and a smooth
Veronese surface. First, we find all the irreducible components of the two intersections, and
describe the geometry of the generic elements in terms of the intersection of P with the other
surface. Then we consider the problem of rationality.
Abstract: TBA
Abstract: TBA
The conference "WINTER MEETING IN ALGEBRA AND GEOMETRY 2025" is a Winter meeting in Algebra and
Geometry in Rome that takes place in the rione Monti.
More info will be displayed in the offical webpage later.
Info
The seminar is usually held every Thursday at 14:15-15:45 in Aula M1. This year seminars will be held in person but you can email one of the organizers if you are interested in attending remotely (through the platform Microsoft Teams).
The seminar is organized by Luca Schaffler and Amos Turchet and maintained by the Geometry Group and the Number Theory Group of the Department of Mathematics and Physics at the Roma Tre University.
We acknowledge the support of the grants PRIN2022: Moduli spaces and birational geometry and PRIN2022: Semiabelian varieties, Galois representations and related Diophantine problems, the 'Programma per Giovani Ricercatori "Rita Levi Montalcini"' and the support of the Department of Mathematics and Physics at Roma Tre University.
