Determinantal schemes and Buchsbaum-Rim sheaves

Abstract

Let $\varphi$ be a generically surjective morphism between direct sums of line bundles on $\proj{n}$ and assume that the degeneracy locus, $X$, of $\varphi$ has the expected codimension. We call $B_{\varphi }=\ker \varphi$ a (first) Buchsbaum-Rim sheaf and we call $X$ a standard determinantal scheme. Viewing $\varphi$ as a matrix (after choosing bases), we say that $X$ is good if one can delete a generalized row from $\varphi$ and have the maximal minors of the resulting submatrix define a scheme of the expected codimension. In this paper we give several characterizations of good determinantal schemes. In particular, it is shown that being a good determinantal scheme of codimension $r+1$ is equivalent to being the zero-locus of a regular section of the dual of a first Buchsbaum-Rim sheaf of rank $r+1$. It is also equivalent to being standard determinantal and locally a complete intersection outside a subscheme $Y\subset X$ of codimension $r+2$. Furthermore, for any good determinantal subscheme $X$ of codimension $r+1$ there is a good determinantal subscheme $S$ codimension $r$ such that $X$ sits in $S$ in a nice way. This leads to several generalizations of a theorem of Kreuzer. For example, we show that for a zeroscheme $X$ in $\proj{3}$, being good determinantal is equivalent to the existence of an arithmetically Cohen-Macaulay curve $S$, which is a local complete intersection, such that $X$ is a subcanonical Cartier divisor on $S$.

20 pages, LaTeX

Cited by in corpus (1)

• Buchsbaum-Rim sheaves and their multiple sections