Conference Agenda

This minisymposium highlights the use of computation inside algebraic geometry. Computations enter algebraic geometry in several different ways including numerical strategies, symbolic calculations, experimentation, and simply as a fundamental conceptual tool. Our speakers will showcase many of these aspects together with some applications.

The computation of the semigroup (or even the cone) of all effective divisor classes on a Calabi–Yau hypersurface of a toric variety is an important open problem, whose solution would have a number of applications in theoretical physics. This is a difficult computation, with no known effective algorithms. We present computational tools and algorithms for investigating this and related problems, and describe some results in this direction.

Numerical Godeaux surfaces provide the first case in the geography of minimal surfaces of general type. By work of Miyaoka and Reid it is known that the torsion group of such a surface is cyclic of order at most 5, a full classification has been given for the cases where this order is 3,4, or 5. In my talk, I will discuss recent progress by Isabel Stenger towards the classification of numerical Godeaux surfaces with a trivial torsion group. Following a suggestion by Frank-Olaf Schreyer, the starting point of Stenger's work is a syzygy-type approach to the study of the canonical ring of such a surface. Particular attention is paid to the hyperelliptic curves arising in the fibration induced by the bicanonical system.

In a joint work with Aldo Conca, we look at what should correspond to Macaulay’s Theorem and Fröberg’s Conjecture for the Hilbert function of subalgebras of standard graded polynomial rings. Upper bounds correspond to generic forms and lower bounds correspond to strongly stable monomial ideals.

In this talk, the concept of singular value decomposition of complexes will introduced and applied to the computation of syzygies.