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Let $Y$ be a quartic double solid and $E$ be an rank two instanton bundle on $Y$. By Serre's correspondence, it is not hard to show that $E$ fits into the following short exact sequence $0\rightarrow\mathcal{O}_Y\rightarrow E(H)\rightarrow I_D(2H)\rightarrow 0,$ where $D$ is the zero locus of general section of $E(H)$, which is a smooth elliptic curve of degree 4.

Denote by $\tau$ the geometric involution of $Y$ induced by double cover. If $\tau(E)\cong E$, do we have $D=\tau(D)?$ The other direction is easy: if $\tau(D)=D$,then $\tau(E)\cong E$, since $H^2(I_D)=k$.

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    $\begingroup$ The isomorphism $\tau^*E \cong E$ induces an involution $\tau^*\colon H^0(E)\to H^0(E)$. The zero locus $D$ is $\tau$-invariant if the corresponding section of $H^0(E)$ is either invariant or anti-invariant. So a general $D$ will not be invariant but you can find one that is. $\endgroup$
    – Daniele A
    Commented Mar 5, 2023 at 19:19

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