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Kim
Kim
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Let $V$ be a $\mathbf{C}$-vector space of dimension $n$, and consider the Grassmannian $G:=Gr(2, V)$ of 2-dim subspaces of $V$. Then we have the tautological subbundle $E\subset V\otimes \mathcal{O}_G$ and the quotient bundle $V\otimes \mathcal{O}_G\twoheadrightarrow F$.

Classically, we know that the zero locus of a non-zero global section of $E^{\vee}$, if non-empty, is $G\cap \mathbf{P}(\wedge^2 W)$, where $W\subset V$ is a codimension one subspace and the intersection is taken under the Plucker embedding $G\hookrightarrow \mathbf{P}(\wedge^2 V)$. Similarly, if we use the natural identification between $Gr(2, V)$ and $Gr(3, V^{\vee})$, then one can see that the non-empty zero locus of a non-zero global section of $F$ is $G\cap \mathbf{P}(W_1\wedge W)$, where $W\subset V$ is a codimension one subspace and $W_1$ is a 1-dim subspace.

Question: Is there any similar description of the non-empty zero locus of a non-zero global section of $\wedge^2 F$?

When $\dim V=5$, I guess it is $G\cap \mathbf{P}(W_2\wedge V)$ (or a linear section of $G\cap \mathbf{P}(W_2\wedge V)$?) according to some lemmata in papers (without proofs), where $W_2$ is a 2-dim subspace of $V$, but I'm not sure how to get this.

Let $V$ be a $\mathbf{C}$-vector space of dimension $n$, and consider the Grassmannian $G:=Gr(2, V)$ of 2-dim subspaces of $V$. Then we have the tautological subbundle $E\subset V\otimes \mathcal{O}_G$ and the quotient bundle $V\otimes \mathcal{O}_G\twoheadrightarrow F$.

Classically, we know that the zero locus of a non-zero global section of $E^{\vee}$, if non-empty, is $G\cap \mathbf{P}(\wedge^2 W)$, where $W\subset V$ is a codimension one subspace and the intersection is taken under the Plucker embedding $G\hookrightarrow \mathbf{P}(\wedge^2 V)$. Similarly, if we use the natural identification between $Gr(2, V)$ and $Gr(3, V^{\vee})$, then one can see that the non-empty zero locus of a non-zero global section of $F$ is $G\cap \mathbf{P}(W_1\wedge W)$, where $W\subset V$ is a codimension one subspace and $W_1$ is a 1-dim subspace.

Question: Is there any similar description of the non-empty zero locus of a non-zero global section of $\wedge^2 F$?

When $\dim V=5$, I guess it is $G\cap \mathbf{P}(W_2\wedge V)$ according to some lemmata in papers (without proofs), where $W_2$ is a 2-dim subspace of $V$, but I'm not sure how to get this.

Let $V$ be a $\mathbf{C}$-vector space of dimension $n$, and consider the Grassmannian $G:=Gr(2, V)$ of 2-dim subspaces of $V$. Then we have the tautological subbundle $E\subset V\otimes \mathcal{O}_G$ and the quotient bundle $V\otimes \mathcal{O}_G\twoheadrightarrow F$.

Classically, we know that the zero locus of a non-zero global section of $E^{\vee}$, if non-empty, is $G\cap \mathbf{P}(\wedge^2 W)$, where $W\subset V$ is a codimension one subspace and the intersection is taken under the Plucker embedding $G\hookrightarrow \mathbf{P}(\wedge^2 V)$. Similarly, if we use the natural identification between $Gr(2, V)$ and $Gr(3, V^{\vee})$, then one can see that the non-empty zero locus of a non-zero global section of $F$ is $G\cap \mathbf{P}(W_1\wedge W)$, where $W\subset V$ is a codimension one subspace and $W_1$ is a 1-dim subspace.

Question: Is there any similar description of the non-empty zero locus of a non-zero global section of $\wedge^2 F$?

When $\dim V=5$, I guess it is $G\cap \mathbf{P}(W_2\wedge V)$ (or a linear section of $G\cap \mathbf{P}(W_2\wedge V)$?) according to some lemmata in papers (without proofs), where $W_2$ is a 2-dim subspace of $V$, but I'm not sure how to get this.

Source Link
Kim
Kim
  • 565
  • 2
  • 7

Zero loci of sections of wedge product of bundles

Let $V$ be a $\mathbf{C}$-vector space of dimension $n$, and consider the Grassmannian $G:=Gr(2, V)$ of 2-dim subspaces of $V$. Then we have the tautological subbundle $E\subset V\otimes \mathcal{O}_G$ and the quotient bundle $V\otimes \mathcal{O}_G\twoheadrightarrow F$.

Classically, we know that the zero locus of a non-zero global section of $E^{\vee}$, if non-empty, is $G\cap \mathbf{P}(\wedge^2 W)$, where $W\subset V$ is a codimension one subspace and the intersection is taken under the Plucker embedding $G\hookrightarrow \mathbf{P}(\wedge^2 V)$. Similarly, if we use the natural identification between $Gr(2, V)$ and $Gr(3, V^{\vee})$, then one can see that the non-empty zero locus of a non-zero global section of $F$ is $G\cap \mathbf{P}(W_1\wedge W)$, where $W\subset V$ is a codimension one subspace and $W_1$ is a 1-dim subspace.

Question: Is there any similar description of the non-empty zero locus of a non-zero global section of $\wedge^2 F$?

When $\dim V=5$, I guess it is $G\cap \mathbf{P}(W_2\wedge V)$ according to some lemmata in papers (without proofs), where $W_2$ is a 2-dim subspace of $V$, but I'm not sure how to get this.