Small schemes and varieties of minimal degree
| dc.creator | Eisenbud, David | |
| dc.creator | Green, Mark | |
| dc.creator | Hulek, Klaus | |
| dc.creator | Popescu, Sorin | |
| dc.date | 2004-04-28 | |
| dc.date | 2004-11-15 | |
| dc.date.accessioned | 2026-07-07T05:07:47Z | |
| dc.date.available | 2026-07-07T05:07:47Z | |
| dc.description | We prove that if X is any 2-regular projective scheme (in the sense of Castelnuovo-Mumford) then X is "small". This means that if L is a linear space and Y:= L\cap X is finite, then Y is "linearly independent" in the sense that the dimension of the linear span of Y is 1+deg Y. The converse is true and well-known for finite schemes, but false in general. The main result of this paper is that the converse, "small implies 2-regular", is also true for reduced projective schemes (algebraic sets). This is proven by means of a delicate geometric analysis, leading to a complete classification: we show that the components of a small algebraic set are varieties of minimal degree, meeting in a particularly simple way. From the classification one can show that if X is 2-regular, then so is X_{red}, and so also is the projection of X from any point of X. Our results extend the Del Pezzo-Bertini classification of varieties of minimal degree, the characterization of these as the varieties of regularity 2 by Eisenbud-Goto, and the construction of 2-regular square-free monomial ideals by Fröberg. | |
| dc.description | 27 pages, Plain TeX + diagrams.tex, minor revisions, updated abstract + title | |
| dc.identifier | https://arxiv.org/abs/math/0404517 | |
| dc.identifier | http://arxiv.org/abs/math/0404517 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/70999 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Commutative Algebra | |
| dc.subject | Combinatorics | |
| dc.subject | 14N05, 14N25, 13D02, 14M20 | |
| dc.title | Small schemes and varieties of minimal degree | |
| dc.type | text |