Abstract. Let R = K[x0, . . . , xn] be the polynomial ring over a field K in n + 1 variables and I be a homogeneous ideal of R. In this talk we discuss a way to analyse Hilbert schemes over the quotient ring R/I using a computer algebra system. The tools that we develop are based on the theory of marked bases over quasi-stable ideals, together with their properties and functorial features. However, the different setting that we consider presents new problems to solve. We apply our tools to achieve two different results under the hypothesis that I is a monomial quasi-stable ideal. The first application produces an explicit open cover of the Hilbert scheme over X when the quotient ring is Cohen-Macaulay over an infinite field. Together with relative marked bases, we use suitable general changes of variables which preserve the structure of the quasi-stable ideal. The second application gives examples of both smooth and singular lex-points in MacaulayLex quotient rings. In particular, computational evidences encourage to think that the lex-point is singular in Clements-Lindström rings. Both these applications take benefits from the fact that the use of marked bases allows the construction of some open subschemes in the Hilbert scheme over X. This is a joint work with Cristina Bertone (University of Turin), Matthias Orth (University of Kassel) and Werner M. Seiler (University of Kassel).
Before this lecture, from 4.45 pm, there will again be coffee and tea in room 1404.
