Let $X$ be a smooth irreducible complex variety of dimension $n \ge 6$. Let $E$ be a globally generated rank $r \ge 2$ vector bundle on $X$ and let $\varphi : {\mathcal O}_X^{\oplus (r-1)} \to E$ be a general morphism. Let $Y = D_{r-2}(\varphi)$ be the degeneracy locus and assume that $Y \ne \emptyset$. Are there examples where $Y$ is not a locally complete intersection?
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2 Answers
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Let $X = \mathbb{P}^6$, $r = 3$, and $E = \mathcal{O}(1)^{\oplus 3}$. Then a general $\varphi$ is given by a matrix $$ \varphi = \begin{pmatrix} x_1 & x_2 \\ x_3 & x_4 \\ x_5 & x_6 \end{pmatrix}, $$ where $(x_0:x_1:\dots:x_6)$ are the homogeneous coordinates, and $$ Y = \{ x_1x_4 - x_2x_3 = x_1x_6 - x_2x_5 = x_3x_6 - x_4x_5 = 0 \} $$ is the cone with vertex $P = (1:0:\dots:0)$ over the Segre variety $\mathbb{P}^1 \times \mathbb{P}^2$, in particular, it is not a local complete intersection at $P$.
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$\begingroup$ How about the case where $X \subset \mathbb P^N$ and $E$ is globally generated and initialized, that is $H^0(E(-1))=0$? I wonder if there are still examples. $\endgroup$– CobCommented Nov 16 at 21:31
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$\begingroup$ There is work of Goto about when determinantal varieties are local complete intersections (and the conclusion is that they are almost never local complete intersections). $\endgroup$ Commented Nov 16 at 22:36
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$\begingroup$ @Cob: If you want an example with $H^0(E(-1)) = 0$, take $X = \mathrm{Gr}(2,5)$ and $E$ the quotient bundle. It s a bit more hard to show, but in this case $Y$ is the same cone over the Segre variety. $\endgroup$– SashaCommented Nov 17 at 8:21
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$\begingroup$ In fact, Goto proves that determinantal loci are typically not even Gorenstein, much less local complete intersections. Here is a link: projecteuclid.org/journals/kyoto-journal-of-mathematics/… $\endgroup$ Commented Nov 17 at 12:43
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I am just posting my comments as one further answer. Every locally complete intersection scheme is Gorenstein (and Cohen-Macaulay). Is a generic determinantal variety Gorenstein? For morphisms from a rank $e$ locally free sheaf to a rank $f$ locally free sheaf and the generic determinantal variety where the rank of the morphism is $\leq r$, Goto proves that the generic determinantal variety is not Gorenstein if both $e\neq f$ and $r\neq 0$, i.e., if it is Gorenstein then either $e$ equals $f$ or $r$ equals $0$ (or both). Also, for $r=1$ and $e=f\leq 5$, Goto proves that the generic determinantal variety is Gorenstein. Here is the reference.
Goto, Shiro
When do the determinantal ideals define Gorenstein rings?.
Sci. Rep. Tokyo Kyoiku Daigaku Sect. A12(1974), 129–145.
https://www.jstor.org/stable/43698822