Dimension of linear systems: a combinatorial and differential approach

1997/09/29 by Laurent Evain, L. Evain, Evain, L.
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #alg-geom #math.AG

paper · pdf · doi:10.48550/arxiv.alg-geom/9709032

17 pages, in french, also available at http://193.49.162.129/~evain/home.html

arxiv created 1997/09/29 · openalex publication_date 1997/09/29 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give upper-bounds for the dimension of some linear systems. The theorem improves the differential Horace method introduced by Alexander-Hirschowitz, and was conjectured by Simpson. Possible applications are the calculus of the dimension of linear systems of hypersurfaces in a projective space \PPⁿ with generically prescribed singularities, and the calculus of collisions of fat points in \PP². These applications will be treated independently but a simple example in the introduction explains how the theorem will be used.

Related

• New effective bounds on the dimension of a linear system in \mathbb P²
• New methods for determining speciality of linear systems based at fat points in Pⁿ
• Linear systems on the blow-up of (P¹)ⁿ
• Linear Systems of Plane Curves with Base Points of Equal Multiplicity
• The Dimension of Quasi-Homogeneous Linear Systems With Multiplicity Four