Asian Journal of Mathematics

Fano Threefolds of Genus 6

Dmitry Logachev

Full-text: Open access

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).

Article information

Source
Asian J. Math. Volume 16, Number 3 (2012), 515-560.

Dates
First available in Project Euclid: 23 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.ajm/1353696020

Mathematical Reviews number (MathSciNet)
MR2989233

Zentralblatt MATH identifier
1263.14040

Subjects
Primary: 14J30: $3$-folds [See also 32Q25] 14J45: Fano varieties
Secondary: 14J25: Special surfaces {For Hilbert modular surfaces, see 14G35} 14C30: Transcendental methods, Hodge theory [See also 14D07, 32G20, 32J25, 32S35], Hodge conjecture

Keywords
Fano threefolds Fano surfaces middle Jacobian tangent bundle theorem global Torelli theorem

Citation

Logachev, Dmitry. Fano Threefolds of Genus 6. Asian J. Math. 16 (2012), no. 3, 515--560.https://projecteuclid.org/euclid.ajm/1353696020


Export citation