Conference Agenda

In this minisymposium, titled "Syzygies and applications to geometry”, we will focus on the striking results and applications that the study of syzygies provides in algebraic geometry, in a wide sense. Topics should include but are not limited to the study of rational and birational maps, singularities, residual intersections and the defining equations of blow-up algebras. We plan to focus on recent progress in this area that result in explicit and effective computations to detect certain geometrical property or invariant. Applications to geometric modeling are very welcome.

Given F=V(f) a reduced projective hypersurface defined by a homogeneous polynomial f in several variables, the gradient of f defines a rational map P_f between two projective spaces of the same dimension, called polar map of F. In general, it is a problem to describe all the homaloidal hypersurfaces, that is the hypersurfaces F=V(f) such that P_f is birational (i.e. P_f is an isomorphism between two Zariski opens) which aims to distinguish specific singular locus of the projective hypersurfaces. The classification of reduced homaloidal complex curves (i.e. when the base field is the complex field) was established by I.V.Dolgachev. It is formed by the smooth conics, the unions of three general lines and the unions of a smooth conics with one of its tangent. When the base field has characteristic p>2, the three curves in Dolgachev's classification are still homaloidal and a problem becomes to establish if they are the only ones. In this talk, I will explain how this question can be related to an analysis of the syzygies of the jacobian ideal of the hypersurfaces and I will show an explicit example of a homaloidal curve of degree 5 if p=3. This can be viewed algebraically as a study of the difference between the Rees and the symmetric algebra of the jacobian ideal or, equivalently, as a study of the variations of the Milnor number of the curves with respect to reduction modulo p.

What kind of reduced monomial schemes can be obtained as a Gröbner degeneration of a smooth projective variety? Emanuela De Negri, Matteo Varbaro and myself conjecture that the answer is: Only Stanley-Reisner schemes associated to acyclic Cohen-Macaulay simplicial complexes. This would imply in particular, that only curves of genus zero have such a degeneration. We proved this conjecture for degrevlex orders, for elliptic curves over real number fields, for boundaries of cross-polytopes, and for leafless graphs. Consequences for rational and F-rational singularities of algebras with straightening laws will also be discussed.

Two numerical semigroups can be glued to obtain another numerical semigroup in higher embedding dimension. This concept was originally introduced to classify numerical semigroups that are complete intersections and it was later generalized to arbitrary numerical semigroups and semigroups in higher dimension. In this talk, we will construct the syzygies of the semigroup ring k[C] of a semigroup C obtained by gluing two semigroups A and B in terms of the syzygies of k[A] and k[B]. This will provide formulas for several invariants like Betti numbers, projective dimension and Hilbert series. We will use our construction to show that gluing two semigroups in higher dimension is not as easy as in the numerical case.

This is a joint work with Hema Srinivasan (Missouri University, USA).