The geometry of the Fano surface of the double cover of $P^3$ branched in a quartic

This paper gives a computation of the irregularity of the Fano surface $\mathscr F$ of lines on the double cover $X\to P^3$ branched in a quartic. A tangent bundle theorem is proved for $\mathscr F$, from which it follows that $\mathscr F$ determines $X$ uniquely. It is shown that the Abel–Jacobi map $a\colon\operatorname{Alb}(\mathscr F)\to J_3(X)$ is an isogeny.
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