# Castelnuovo-Mumford Regularity of Ideals of Maximal Minors

I have an $m \times 2m$ matrix of linear forms over $\mathbb{C}[x,y,z,w]$. It is of the form $$M = ( x I - A z -B w \mid y I - C z - D w).$$ Here $A,B,C$ and $D$ are $m \times m$ scalar matrices. Let $I$ be the ideal of maximal minors of $M$. I believe that the CM-regularity of this ideal is $m$. However I have no idea how to show it.

This is not a generic determinantal ideal, so many results for them do not apply in this case. However maybe there is a way to specialize and still keep the CM-regularity.

In general I want to show that for matrices of different size the closed subschemes of $\mathbb{P}^3$ defined by the ideals are non-isomorphic. Under certain assumptions the subscheme will be one dimensional.

I'm not very familiar with results about the CM-regularity (outside Eisenbud's "Commutative Algebra"), so a reference of something will be appreciated as well.

Thanks.

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## 2 Answers

Let me first rephrase the question. You consider a morphism of vector bundles $$O(-1)^{2m} \to O^{m}$$ on $P^3$ and ask about its degeneration scheme. The standard way to describe it as follows. Consider the product $P^3 \times P^{m-1}$ (with the second factor being the space of 1-dimensional quotients of $\mathbb{C}^m$. Then on the product consider the composition $$O(-1,0)^{2m} \to O^{m} \to O(0,1),$$ where the second morphism is tautological. Let $Z$ be its zero locus. Then the degeneration scheme is the image of $Z$. In particular, if $m > 2$ then with generic choice of matrices $Z$ is empty (and hence so is the degeneration scheme). So, from now on I assume $m \le 2$ and the matrices are general.

Under these assumptions the scheme $Z$ has a Koszul resolution $$\dots \to O(-2,-2)^{\binom{2m}{2}} \to O(-1,-1)^m \to O \to O_Z \to 0.$$ Pushing it forward to $P^3$ and using the cohomology of line bundles on $P^{m-1}$, one gets the following resolution for the degeneration scheme $D \subset P^3$: $$\dots \to O(-1-m)^{\binom{2m}{m+1}} \to O(-m)^{\binom{2m}{m}} \to O \to O_D \to 0.$$ So, the CM-regularity of the ideal is $I$.

Of course, the case $m \le 2$ is not very interesting, but if you replace $P^3$ with a projective space of higher dimension, and then the same argument can be used for higher $m$ as well.

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One possible explanation for what you are claiming is that the ideal of m-minors is equal to the m-th power of the maximal ideal (x,y,z,w). Did you check if this is the case?

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