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Sam Hopkins
Sam Hopkins
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Consider the set $$\\\{ (A,B) \in \mathbb{P}^{n\times n-1} \times \mathbb{P}^{n\times n -1} : \text{im}(A) \subseteq \text{im}(B)\\\}.$$$$\\\{ (A,B) \in \mathbb{P}^{n\times n-1} \times \mathbb{P}^{n\times n -1} : \text{im}(A) \subseteq \text{im}(B)\}.$$ That is, this is the set of pairs of square matrices $(A,B)$ so that the image of $A$ is contained in the image of $B$. Is this Zariski closed? I would be happy if this were at least true over an algebraically closed field.

I tried to write down explicit equations for this, but my first attempt would involved using determinants of submatrices of $A$, and would fail if $A$ was singular. I had thought that this would follow from the projection theorem for for projective varieties, but am unsure.

Consider the set $$\\\{ (A,B) \in \mathbb{P}^{n\times n-1} \times \mathbb{P}^{n\times n -1} : \text{im}(A) \subseteq \text{im}(B)\\\}.$$ That is, this is the set of pairs of square matrices $(A,B)$ so that the image of $A$ is contained in the image of $B$. Is this Zariski closed? I would be happy if this were at least true over an algebraically closed field.

I tried to write down explicit equations for this, but my first attempt would involved using determinants of submatrices of $A$, and would fail if $A$ was singular. I had thought that this would follow from the projection theorem for for projective varieties, but am unsure.

Consider the set $$\\\{ (A,B) \in \mathbb{P}^{n\times n-1} \times \mathbb{P}^{n\times n -1} : \text{im}(A) \subseteq \text{im}(B)\}.$$ That is, this is the set of pairs of square matrices $(A,B)$ so that the image of $A$ is contained in the image of $B$. Is this Zariski closed? I would be happy if this were at least true over an algebraically closed field.

I tried to write down explicit equations for this, but my first attempt would involved using determinants of submatrices of $A$, and would fail if $A$ was singular. I had thought that this would follow from the projection theorem for for projective varieties, but am unsure.

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Quizzical
Quizzical
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Is containment of images of linear maps Zariski closed?

Consider the set $$\\\{ (A,B) \in \mathbb{P}^{n\times n-1} \times \mathbb{P}^{n\times n -1} : \text{im}(A) \subseteq \text{im}(B)\\\}.$$ That is, this is the set of pairs of square matrices $(A,B)$ so that the image of $A$ is contained in the image of $B$. Is this Zariski closed? I would be happy if this were at least true over an algebraically closed field.

I tried to write down explicit equations for this, but my first attempt would involved using determinants of submatrices of $A$, and would fail if $A$ was singular. I had thought that this would follow from the projection theorem for for projective varieties, but am unsure.