Research interests

I am presently interested in commutative algebra and related topics in effective algebraic geometry (i.e. computational algebra) and geometric modeling.

1. Rational maps and birational maps : The study of rational maps is of theoretical interest in algebraic geometry and commutative algebra, and of practical importance in geometric modeling. My research focus on rational maps in low dimension, typically parameterizations of curves and surfaces embedded in the projective space of dimension 3, but also dominant rational maps in dimension two and three. The two main objectives amount to unravel geometric properties of these rational maps from the syzygies of their projective coordinates. The first one aims at extending and generalizing the determination of the closed image of a rational map, as well as its geometric features, whereas the second one will focus on the study of dominant rational maps, in particular on the characterization of those that are generically one-to-one.
2. Residual intersections: The concept of residual intersection was introduced by Artin and Nagata in 1972, as a generalization of linkage; it is more ubiquitous, but also harder to understand. I am interested in studying the Cohen-Macaulayness and the canonical module of residual intersections.
3. Koszul homologies of powers of the maximal ideal: A classical problem in algebraic geometry and commutative algebra is the study of the equations defining projective varieties and of their syzygies. My approach is based on the study of the Koszul homologies of powers of the maximal ideal in order to find a lower bound for Green–Lazarsfeld index .
4. The weak and/or strong Lefschetz property of some artinian algebras, in particular, codimesion three artinian Gorenstein algebras and complete intersections.