Fano threefolds of genus 6

摘要

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) ? 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 Φ: Alb F(X) → 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. im Φ(F(X)) ? J ~3(X) is algebraically equivalent to 2Θ8/8! where Θ ? J ~3(X) is a Poincaré divisor (a sketch of the proof).