Báo cáo viên: Prof. Martin Kreuzer (University of Passau)
Thời gian: 14h, ngày 27/3/2025
Địa điểm: Báo cáo thực hiện trực tiếp tại: Phòng 612 nhà A6 - Viện Toán học
và Online (Join Zoom Meeting) tại link: https://zoom.us/j/99636681387?pwd=0WscBnehOJig68SqctGluVuA3RwraE.1
Tóm tắt: One of the key features of Algebraic Geometry is the existence of moduli spaces, i.e., of schemes whose closed points correspond to certain types of algebraic varieties or schemes. An intensely studied case is the Hilbert scheme Hilb^mu(mathbb{P}^n_K) parametrizing 0-dimensional subschemes of a fixed projective space over a field K. Performing explicit computer calculations for these schemes has been notoriously difficult, because the presentations of their coordinate rings provided by Grothendieck's construction are hard to make explicit and involve large numbers of indeterminates and defining equations.
Here border basis schemes come to the rescue. For an order ideal mathcal{O} of terms, i.e., for a divisor-closed finite set of terms, the border basis scheme mathbb{B}_mathcal{O} parametrizes all 0-dimensional affine schemes for which the terms in mathcal{O} define a K-vector space basis of their coordinate ring. These schemes form an open covering of the Hilbert scheme and have explicitly describable, well-manageable defining equations.
After recalling the construction and some basic properties of border basis schemes, we survey some recent joint work with Lorenzo Robbiano (Genova) and Le Ngoc Long (Hue) concerning their computational aspects. We consider important subschemes of mathbb{B}_mathcal{O} such as the homogeneous border basis scheme, the maxdeg border basis scheme, and various subschemes parametrizing properties such as being locally Gorenstein, strictly Gorenstein, strict complete intersections, having the Cayley-Bacharach property, etc.
The last topic is a new technique, called Z-separating embeddings, for re-embedding schemes from high-dimensional spaces into lower-dimensional spaces which avoids the potentially costly calculation of Gr"obner bases and allows us, for instance, to prove that certain border bases schemes are isomorphic to affine spaces and some are not. | 
