Beitr\"age zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 36 (1995), No. 1
Vector Spaces of Matrices of Low Rank
and Vector Bundles on Projectic Spaces:
an addendum to a paper by Eisenbud and Harris
E. Ballico
Here we give a quick affirmative answer to a linear algebra question raised in
D. Eisenbud and J. Harris, Vector spaces of matrices of low rank,
Adv. in Math. 70 (1988), 135-155. In the quoted paper methods coming from
algebraic geometry (essentially the theory of vector bundles and reflexive
sheaves on $\P^N$) were shown to be powerful tools for the study of vector
spaces of matrices. Fix a positive integer $k$. Let $V$ and $W$ be vector
spaces; set $v:= \dim(V),\ w:= \dim(W),\ s:= \min(v,w)$. Let $M\subseteq
Hom(V,W)$ be a vector space of linear maps such that the general $f\in M$ has
rank $k$. Here we show essentially that if $k$ is fixed and ``$M$ does not
come from a simpler situation'', then $s$ cannot be arbitrarily large. With
the notations of the quoted paper we show the existence of $s_0(k)$ such that
if $s\ge s_0(k)$, then $M$ is not strongly indecomposable.