 # 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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