Fano Threefolds of Genus 6

Abstract

Ideas and methods of Clemens C. H., Griffiths Ph. The intermediate Jacobian of a cubic threefold are applied to a Fano threefold $X$ of genus 6 — intersection of $G(2, 5) \subset P^9$ with $P^7$ and a quadric. Main results:

1. The Fano surface $F(X)$ of $X$ is smooth and irreducible. Hodge numbers and some other invariants of $F(X)$ are calculated.

2. Tangent bundle theorem for $X$ is proved, and its geometric interpretation is given. It is shown that $F(X)$ defines $X$ uniquely.

3. The Abel-Jacobi map $\Phi : \operatorname{Alb} F(X) \to J^3(X)$ is an isogeny.

4. As a necessary step of calculation of $h^{1,0}(F(X))$ we describe a special intersection of 3 quadrics in $P^6$ (having 1 double point) whose Hesse curve is a smooth plane curve of degree 6.

5. $\operatorname{im} \Phi(F(X)) \subset J^3(X)$ is algebraically equivalent to $\frac{2\Theta^8}{8!}$ where $\Theta \subset J^3(X)$ is a Poincaré divisor (a sketch of the proof).