I am fascinated by the study of varieties defined over an arithmetically rich field, such as the one of rational numbers. The arithmetic complexity of their subvarieties is described by suitable height functions, which can be defined with great freedom in the framework of adelic Arakelov geometry.

Heights represent an efficient vehicle of the deep interactions between geometry and arithmetics: they appear for instance in Faltings's proof of the Mordell conjecture, exhibiting the influence of the geometrical properties of a curve on the number of its rational points, and they determine the geometric behaviour of certain sequences of "arithmetically simple subvarieties" in the setting of equidistribution theory.

Conceptually, heights can be recovered as the arithmetic analogue of the geometric degree through Arakelov geometry. This subject, in which algebraic geometry of schemes over the integers gets mixed with complex differential geometry, is an arithmetic version of classical intersection theory, and it allows to formulate analogues of geometric results and problems, including the arithmetic standard conjectures.

In this framework, I am particularly interested in the interplay between arithmetic and convex geometry: there exists a series of toolkits and constructions for passing from on side to the other, including toric and tropical geometry. I am mainly excited by the idea of using these tools to recover arithmetic information about varieties by the study of their convex counterparts.

• (with Martín Sombra) Heights of complete intersections in toric varieties, preprint available at arXiv:2412.16308. Read more  
  The height of a toric variety and that of its hypersurfaces can be expressed in convex-analytic terms as an adelic sum of mixed integrals of their roof functions and duals of their Ronkin functions. Here we extend these results to the 2-codimensional situation by presenting a limit formula predicting the typical height of the intersection of two hypersurfaces on a toric variety. More precisely, we prove that the height of the intersection cycle of two effective divisors translated by a strict sequence of torsion points converges to an adelic sum of mixed integrals of roof and duals of Ronkin functions. This partially confirms a previous conjecture of the authors about the average height of families of complete intersections in toric varieties.  
• (with Martín Sombra) Limit heights and special values of the Riemann zeta function, Journal of Experimental Mathematics 1 (2025), no. 2, 322 — 374 (arXiv version). Read more  
  We study the distribution of the height of the intersection between the projective line defined by the linear polynomial x0+x1+x2 and its translate by a torsion point. We show that for a strict sequence of torsion points, the corresponding heights converge to a real number that is a rational multiple of a quotient of special values of the Riemann zeta function. We also determine the range of these heights, characterize the extremal cases, and study their limit for sequences of torsion points that are strict in proper algebraic subgroups. In addition, we interpret our main result from the viewpoint of Arakelov geometry, showing that for a strict sequence of torsion points the limit of the corresponding heights coincides with an Arakelov height of the cycle of the projective plane over the integers defined by the same linear polynomial. This is a particular case of a conjectural asymptotic version of the arithmetic Bézout theorem. Using the interplay between arithmetic and convex objects from the Arakelov geometry of toric varieties, we show that this Arakelov height can be expressed as the mean of a piecewise linear function on the amoeba of the projective line, which in turn can be computed as the aforementioned real number.  
• (with Paolo Dolce) Numerical equivalence of ℝ-divisors and Shioda-Tate formula for arithmetic varieties, Journal für die reine und angewandte Mathematik 784 (2022), 131 — 154 (arXiv version). Read more  
  Let X be an arithmetic variety over the ring of integers of a number field K, with smooth generic fiber X_K. We give a formula that relates the dimension of the first Arakelov-Chow vector space of X with the Mordell-Weil rank of the Albanese variety of X_K and the rank of the Néron-Severi group of X_K. This is a higher dimensional and arithmetic version of the classical Shioda-Tate formula for elliptic surfaces. Such analogy is strengthened by the fact that we show that the numerically trivial arithmetic ℝ-divisors on X are exactly the linear combinations of principal ones. This result is equivalent to the non-degeneracy of the arithmetic intersection pairing in the argument of divisors, partially confirming a conjecture by Gillet and Soulé.  
• (with César Martínez) Higher dimensional essential minima and equidistribution of cycles, Annales de l'Institut Fourier 72 (2022), no. 4, 1329 — 1377 (arXiv version). Read more  
  The essential minimum and equidistribution of small points are two well-established interrelated subjects in arithmetic geometry. However, there is lack of an analogue of essential minimum dealing with higher dimensional subvarieties, and the equidistribution of these is a far less explored topic.
  In this paper, we introduce a new notion of higher dimensional essential minimum and use it to prove equidistribution of generic and small effective cycles. The latter generalizes the previous higher dimensional equidistribution theorems by considering cycles and by allowing more fexibility on the arithmetic datum.  
• Heights of hypersurfaces in toric varieties, Algebra & Number Theory 12 (2018), no. 10, 2403 — 2443 (arXiv version). Read more  
  For a cycle of codimension 1 in a toric variety, its degree with respect to a nef toric divisor can be understood in terms of the mixed volume of the polytopes associated to the divisor and to the cycle. We prove here that an analogous combinatorial formula holds in the arithmetic setting: the global height of a 1-codimensional cycle with respect to a toric divisor equipped with a semipositive toric metric can be expressed in terms of mixed integrals of the v-adic roof functions associated to the metric and the Legendre-Fenchel dual of the v-adic Ronkin function of the Laurent polynomial of the cycle.

• ¿Cuántas raíces de la unidad anulan un polinomio en dos variables?, La Gaceta de la Real Sociedad Matemática Española 26 (2023), 149 — 172 (original Spanish version and English version). Read more  
  In these expository notes, we consider the task of searching for roots of unity which are solutions of a bivariate polynomial with algebraic coefficients. This apparently innocent question will drive us in an enchanting journey involving heights, equidistribution phenomena and famous conjectures in the diophantine realm, namely the former toric Manin—Mumford and Bogomolov conjectures. Rather than on proving new results or giving a detailed survey of the literature, the focus of these pages is on providing an elementary presentation of fundamental tools and techniques in arithmetic and diophantine geometry, and on showing some of their most spectacular applications in a down-to-earth way, with the secret goal of motivating students and nonexperts to dive into this fascinating topic. At the end, we point to several directions that have recently been and still are exciting active research areas.  

• (with Martín Sombra) Computing the height of the intersection of the projective line (x0+x1+x2=0)⊂P2 and its translate by a torsion point, Sagemath Notebook (2022), available online at bit.ly/3GFNAcB. Read more  
  This notebook provides the companion code for the paper "Limit heights and special values of the Riemann zeta function". The main focus is the study of the (canonical) height of the intersection point between the projective plane line (x0+x1+x2=0) and its translates by nontrivial torsion points of the 2-dimensional multiplicative torus. The average of such height values, their asymptotic behaviour, and their concentration around a certain expected average are considered. 
• Equidistribution of algebraic integers having small height, Sagemath Notebook (2022), available online at bit.ly/3jxYgR4. Read more  
  This notebook provides a visualization for Bilu's equidistribution theorem in a special one-dimensional case. More precisely, we provide functions constructing algebraic integers with arbitrarily small height, and graphically show that their Archimedean Galois orbits approach the unit circle in the complex plane.  

Here are some unpublished (and unpublishable) personal notes. They were written to be used mostly by their author; no completeness nor consistency should be expected.

• Valuative description of multiplier ideals (2021). Following lecture notes by S. Boucksom, we explain how to express the multiplier ideal associated to a plurisubharmonic function at a point as a valuative ideal. This description involves generic Lelong numbers and the log-discrepancy of divisors lying on modications of the original manifold.
• The amoeba of a planar line (2020). In these elementary notes, we analytically describe the shape of the archimedean amoeba of the zero set of an affine polynomial in two variables with complex coefficients.
• A note about harmonic functions (2014). The main aim of these few lines is to check by explicit computations that the logarithm of the modulus is a harmonic function on the holed complex plane. This implies in particular Jensen's formula and a relation between the height of an algebraic number and the logarithmic Mahler measure of its minimal polynomial.

• On varieties in algebraic tori and their torsion points, in occasion of a virtual talk in the "Higher Invariants Oberseminar (HIOB)" at Universität Regensburg (April 2021). In this expository talk, motivated by the elementary question of finding roots of unity which solve a two-variables polynomial, we dive into the world of heights, present an equidistribution theorem due to Yu. Bilu, and apply it to prove the former toric Bogomolov conjecture in diophantine geometry.
• Higher essential minima at the service of Arakelov equidistribution, prepared for a virtual talk in the "Oberseminar über Arakelov Theorie" at Universität Regensburg (June 2020). The goal of the presentation was to illustrate to an heterogeneous audience the results obtained in a joint work with César Martínez.
• Heights of cycles in toric varieties, in occasion of a talk given at the "Intercity Seminar on Arakelov Geometry" at the University of Copenhagen (September 2018). Having being given a couple of weeks before my defense, the talk summarises a good part of the results I have obtained during my PhD thesis.
• Cox Rings for a particular class of toric schemes, in occasion of the defense of my master thesis (July 2014). I was too young when I prepared the text and I would now make some changes.
• Funzioni Aritmetiche (in Italian), in occasion of the defense of my bachelor thesis (November 2012).

Here are my bachelor, master and Phd theses.

• Height of cycles in toric varieties, Phd thesis (defended in September 2018), advisors Prof. Martín Sombra and Prof. Alain Yger. Read more  
  I investigate in this work the relation between suitable Arakelov heights of a cycle in a toric variety and the arithmetic features of its defining Laurent polynomials. To this purpose, I associate to a Laurent polynomial certain concave functions which I call Ronkin functions and upper functions. I give upper bounds for the height of a complete intersection in terms of the associated upper functions. For a hypersurfaces, I prove a formula relating its height to the Ronkin function of the associated Laurent polynomial. I conjecture an analogous equality for a suitable average height in higher codimensions and indicate a strategy for the proof of a particular case. In all the treatment, I deal with convex geometrical objects such as polytopes, real Monge-Ampère measures and Legendre-Fenchel duality of concave functions. I suggest an algebraic framework for such a study and deepen the understanding of mixed integrals. 
• Cox Rings for a particular class of toric schemes, master thesis (defended in July 2014), advisor Prof. Alain Yger. Read more  
  This represents my first contact with toric varieties and the text results to be very introductory (and somewhere not optimal even in definitions). The focus is put on the construction of abstract toric varieties on any base ring starting from fans. The central part stresses the relations between the properties of the input (the fan and the ring) and the output (the toric variety) of this construction. Cox's construction and the relation between Cox rings and categorical quotients is briefly presented in the end and stated only for algebraically closed fields. 
• Funzioni Aritmetiche (con particolare attenzione alla funzione di Möbius e ad alcune sue applicazioni) (in Italian), bachelor thesis (defended in November 2012), advisor Prof. Thomas Stefan Weigel. Read more  
  The text is an elementary study of arithmetic functions, i.e. functions defined on the positive integers. They form a commutative ring with respect to pointwise addition and Dirichlet convolution. Here we focus on the Möbius function and on some application of Möbius inversion formula; for instance, the number of monic irreducible polynomials of fixed degree with coefficients in a finite field is computed. The last chapter deals with an introduction to Dirichlet series. Abstracts in Italian and English are available.