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Let $X$ be a degree 12 or degree 16 index one prime Fano threefold. In the paper of Mukai https://arxiv.org/pdf/math/0304303.pdf page 500, Theorem 4 and Theorem 5. He said $X_{12}$ has two ambient spaces: Grassmannian $\mathrm{Gr}(5,10)$ and moduli space of rank 2 stable vector bundles over a genus 7 curve $C$, denote by $\mathcal{U}_C(2)$. This two space have the same dimension 25. (Similar for $X_{16}$, its two ambient spaces are $\mathrm{Gr}(3,6)$ and $\mathcal{U}_{C'}(2)$, both have dimension 9, where genus of $C'$ is 3). Mukai says that the two ambient spaces are dual to each other in the sense that the normal bundle of X inside the two spaces are twisted dual to each other: If $N_1$ is the normal bundle of $X$ in $\mathcal{U}_C(2)$ and $N_2$ is the normal bundle of $X$ in $\mathrm{Gr}(5,10)$, then $N_2\cong N_1\otimes\mathcal{O}_X(H)$$N_2\cong N_1^{\vee}\otimes\mathcal{O}_X(H)$. My first question is How to compute the normal bundle of $X$ inside $\mathcal{U}_C(2)$? Note that $X$ is a Brill-Noether loci of $\mathcal{U}_C(2)$. The second question is that is there more geometric relation between $\mathcal{U}_C(2)$ and $\mathrm{Gr}(5,10)$?(or $\mathcal{U}_{C'}(2)$ and $\mathrm{Gr}(3,6)$).

Let $X$ be a degree 12 or degree 16 index one prime Fano threefold. In the paper of Mukai https://arxiv.org/pdf/math/0304303.pdf page 500, Theorem 4 and Theorem 5. He said $X_{12}$ has two ambient spaces: Grassmannian $\mathrm{Gr}(5,10)$ and moduli space of rank 2 stable vector bundles over a genus 7 curve $C$, denote by $\mathcal{U}_C(2)$. This two space have the same dimension 25. (Similar for $X_{16}$, its two ambient spaces are $\mathrm{Gr}(3,6)$ and $\mathcal{U}_{C'}(2)$, both have dimension 9, where genus of $C'$ is 3). Mukai says that the two ambient spaces are dual to each other in the sense that the normal bundle of X inside the two spaces are twisted dual to each other: If $N_1$ is the normal bundle of $X$ in $\mathcal{U}_C(2)$ and $N_2$ is the normal bundle of $X$ in $\mathrm{Gr}(5,10)$, then $N_2\cong N_1\otimes\mathcal{O}_X(H)$. My first question is How to compute the normal bundle of $X$ inside $\mathcal{U}_C(2)$? Note that $X$ is a Brill-Noether loci of $\mathcal{U}_C(2)$. The second question is that is there more geometric relation between $\mathcal{U}_C(2)$ and $\mathrm{Gr}(5,10)$?(or $\mathcal{U}_{C'}(2)$ and $\mathrm{Gr}(3,6)$).

Let $X$ be a degree 12 or degree 16 index one prime Fano threefold. In the paper of Mukai https://arxiv.org/pdf/math/0304303.pdf page 500, Theorem 4 and Theorem 5. He said $X_{12}$ has two ambient spaces: Grassmannian $\mathrm{Gr}(5,10)$ and moduli space of rank 2 stable vector bundles over a genus 7 curve $C$, denote by $\mathcal{U}_C(2)$. This two space have the same dimension 25. (Similar for $X_{16}$, its two ambient spaces are $\mathrm{Gr}(3,6)$ and $\mathcal{U}_{C'}(2)$, both have dimension 9, where genus of $C'$ is 3). Mukai says that the two ambient spaces are dual to each other in the sense that the normal bundle of X inside the two spaces are twisted dual to each other: If $N_1$ is the normal bundle of $X$ in $\mathcal{U}_C(2)$ and $N_2$ is the normal bundle of $X$ in $\mathrm{Gr}(5,10)$, then $N_2\cong N_1^{\vee}\otimes\mathcal{O}_X(H)$. My first question is How to compute the normal bundle of $X$ inside $\mathcal{U}_C(2)$? Note that $X$ is a Brill-Noether loci of $\mathcal{U}_C(2)$. The second question is that is there more geometric relation between $\mathcal{U}_C(2)$ and $\mathrm{Gr}(5,10)$?(or $\mathcal{U}_{C'}(2)$ and $\mathrm{Gr}(3,6)$).

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asked Nov 27, 2021 at 19:50

user41650

Normal bundle of a Fano threefold as Brill-Noether loci

Let $X$ be a degree 12 or degree 16 index one prime Fano threefold. In the paper of Mukai https://arxiv.org/pdf/math/0304303.pdf page 500, Theorem 4 and Theorem 5. He said $X_{12}$ has two ambient spaces: Grassmannian $\mathrm{Gr}(5,10)$ and moduli space of rank 2 stable vector bundles over a genus 7 curve $C$, denote by $\mathcal{U}_C(2)$. This two space have the same dimension 25. (Similar for $X_{16}$, its two ambient spaces are $\mathrm{Gr}(3,6)$ and $\mathcal{U}_{C'}(2)$, both have dimension 9, where genus of $C'$ is 3). Mukai says that the two ambient spaces are dual to each other in the sense that the normal bundle of X inside the two spaces are twisted dual to each other: If $N_1$ is the normal bundle of $X$ in $\mathcal{U}_C(2)$ and $N_2$ is the normal bundle of $X$ in $\mathrm{Gr}(5,10)$, then $N_2\cong N_1\otimes\mathcal{O}_X(H)$. My first question is How to compute the normal bundle of $X$ inside $\mathcal{U}_C(2)$? Note that $X$ is a Brill-Noether loci of $\mathcal{U}_C(2)$. The second question is that is there more geometric relation between $\mathcal{U}_C(2)$ and $\mathrm{Gr}(5,10)$?(or $\mathcal{U}_{C'}(2)$ and $\mathrm{Gr}(3,6)$).