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Hilbert Functions of Fat Point Schemes

Banff International Research Station for Mathematical Innovation and Discovery

Information on the Hilbert function of a finite set of fat points is relevant in a variety of studies such as interpolation, linear systems of rational varieties, or the weak Lefschetz property. In particular, one is interested in the least degree from which on the Hilbert function is predictable or, equivalently, its Castelnuovo-Mumford regularity. An optimal upper bound, named after Segre, was conjectured by Trung and, independently, by Fatabbi and Lorenzini in 2001. In joint work with Bill Trok, we recently established this conjecture. Furthermore, we derive an alternate regularity bound that improves the Segre bound in some cases, for example, if the support contains a subset of general points. Among the arguments is a new partition result for matroids.

Mathematics; Commutative rings and algebras; Algebraic geometry; Algebraic structures

video/mp4

Author affiliation: University of Kentucky

2017-11-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63602

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/