InterestsI am particularly fascinated by the interplay between Algebraic Geometry and (real) Convex Geometry: there exists a series of toolkits and constructions for passing from on side to the other. Basic examples are Toric Geometry and Tropical Geometry.
I am mainly excited by the idea of recovering information about algebraic and arithmetical properties of varieties by the study of their convex counterparts. This is related to the attempt of understanding which kinds of information survive the passage from the algebro-geometrical world to the convex one.
Publications and preprints
- Heights of hypersurfaces in toric varieties. (arXiv)
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.
Personal notesHere are some unpublished (and unpublishable) personal notes. They were written to be used mostly by myself; no completeness nor consistency should be expected.
- A note about harmonic functions. 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 Mahler measure of its minimal polynomial.
- 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 (e.g. require the torus to be dense in the toric scheme, of course).
- Funzioni Aritmetiche (in Italian), in occasion of the defense of my bachelor thesis (November 2012).
ThesesHere are my bachelor and master theses.
- Cox Rings for a particular class of toric schemes, master thesis (defended in July 2014), advisor Prof. Alain Yger. 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. 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.