Orlando Villamayor

Biography

"My research is oriented towards questions of commutative algebra and algebraic geometry. I have worked particularly on resolution of singularities. Hironaka proved the existence of resolution on a wide class of rings containing a field of characteristic zero. His proof is existential, but for many applications it is natural to ask if there is an explicit manner to resolve the singularities. For instance, if a group acts on a singular scheme it natural to ask if a resolution can be defined so that the group action can be lifted. This motivated my work on constructive (or algorithmic) resolution, and later to some application of this result. One application arises when considering a family of singular schemes, for example a Hilbert scheme of projective varieties. The algorithm of resolution provides a natural stratification of the family into equiresolvable families, in which the different members of these families undergo a similar resolution of singularities. Another application of constructive resolution was the proof of a question of Zariski in relation to his theory of equisingularity. Constructive resolutions shows that singularities undergo similar resolutions along the points of a equisingular stratum. My research is also oriented to the study of rings and schemes of positive characteristic, particularly in the last years. In this setting resolution of singularities remains as an open problem, and there are a number of very natural and exiting questions within this frame. The publications listed in my web page are a clear evidence of the interest of our working group on singularities in positive characteristic."

Research Interest

Commutative algebra and algebraic geometry