## Final Report Summary - GSV (The Geometry of Severi varieties on toric surfaces)

Summary description of the project objectives

My research relates to the area of algebraic geometry, a branch of modern pure mathematics, and it includes some aspects of tropical geometry, a combinatorial piece-wise linear geometry. In this project, I study the geometry of Severi varieties on toric surfaces in arbitrary characteristic. I address four main problems, as follows:

Problem 1 (Dimension problem) Find the dimension of VΣ,L ,g.

Problem 2 (Geometry of curves) Describe the geometry of a general curve C of genus g in |L |, in particular classify the singularities of C.

Problem 3 (Enumeration of curves) Find enumerative formulas for the number of curves of genus g in |L | satisfying certain (linear) constraints, e.g., passing through a collection of given points in general position.

Problem 4 (Irreducibility problem) Find examples of toric surfaces admitting reducible/irreducible Severi varieties V irr Σ,L ,g in positive characteristic. Classify the toric surfaces Σ for which V irr Σ,L ,g are irreducible. Prove the irreducibility of Severi varieties V irr Σ,L ,g on toric surfaces in characteristic zero.

The work performed since the beginning of the project

Transfer of knowledge: I gave two graduate courses on Algebraic Geometry, organized students seminar for several terms, an international workshop, and have been organizing a weekly seminar on Algebraic Geometry and Number Theory for the last five years. I also attended several international conferences, and gave many talks on the topic of the project at the conferences and at several seminars in Brazil, France, Germany, Israel, Switzerland, Turkey, and US.

Research: I have achieved most of the objectives and technical goals of the project with natural deviations due to the development of my research. The main results are described below.

My research relates to the area of algebraic geometry, a branch of modern pure mathematics, and it includes some aspects of tropical geometry, a combinatorial piece-wise linear geometry. In this project, I study the geometry of Severi varieties on toric surfaces in arbitrary characteristic. I address four main problems, as follows:

Problem 1 (Dimension problem) Find the dimension of VΣ,L ,g.

Problem 2 (Geometry of curves) Describe the geometry of a general curve C of genus g in |L |, in particular classify the singularities of C.

Problem 3 (Enumeration of curves) Find enumerative formulas for the number of curves of genus g in |L | satisfying certain (linear) constraints, e.g., passing through a collection of given points in general position.

Problem 4 (Irreducibility problem) Find examples of toric surfaces admitting reducible/irreducible Severi varieties V irr Σ,L ,g in positive characteristic. Classify the toric surfaces Σ for which V irr Σ,L ,g are irreducible. Prove the irreducibility of Severi varieties V irr Σ,L ,g on toric surfaces in characteristic zero.

The work performed since the beginning of the project

Transfer of knowledge: I gave two graduate courses on Algebraic Geometry, organized students seminar for several terms, an international workshop, and have been organizing a weekly seminar on Algebraic Geometry and Number Theory for the last five years. I also attended several international conferences, and gave many talks on the topic of the project at the conferences and at several seminars in Brazil, France, Germany, Israel, Switzerland, Turkey, and US.

Research: I have achieved most of the objectives and technical goals of the project with natural deviations due to the development of my research. The main results are described below.