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ag.algebraic geometry – Image matrix with linear entries in $okay[x_1, ldots, x_q]$ whose $r$-minors are all proportional Answer

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ag.algebraic geometry – Image matrix with linear entries in $okay[x_1, ldots, x_q]$ whose $r$-minors are all proportional

Let $n,m geq 2$, $1 leq r < mathrm{min}(m,n)$, $A_1, ldots, A_q in mathcal{M}_{m,n}(okay)$ (the place $okay$ is a bailiwick, algebraically closed if needful). Assume that the next are happy:

$bullet$ all of the $r occasions r$ minors of $x_1A_1+ ldots + x_qA_q$ are vanishing within the polynomial ring $okay[x_1 ldots, x_q]$,

$bullet$ there exists an $r-1 occasions r-1$ minor of $x_1A_1+ ldots + x_qA_q$ which doesn’t vanish and all $r-1 occasions r-1$ minors of $x_1A_1+ ldots + x_qA_q$ are all proportional to a hard and fast $r-1 occasions r-1$ minor which is integral in $okay[x_1, ldots x_q]$ (i.e. they’re both $0$ or they outline the identical integral hypersurface in $mathbb{P}^{q-1}$).

Can we dedcue that there exists a hard and fast hyperplane $H subset okay^m$ such that:
$$ forall (x_1, ldots, x_q) in okay^q, mathrm{Im}(x_1A_1+ ldots + x_qA_q) subset H ?$$

In illustration $r=2$, the respond is clearly “yes”. In illustration $r=3$, I cerebrate I too have a proof that the respond is optimistic, however it’s fairly hideous. I used to be questioning if the outcome is undoubted for any $r< mathrm{min}(m,n)$.

In the langage of algebraic geometry, I’m asking the next. Let $M$ breathe a $m occasions n$ matrix with linear entries in $okay[x_1, ldots, x_q]$. Assume that the cokernel of $M$ is generically of strictly optimistic rank on $mathbb{P}^{q-1}$ and that the degeneracy locus of $M$ is an integral hypersurface in $mathbb{P}^{q-1}$. Do we essentially have $H^0(mathrm{coker}(M)^*) neq 0$?

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