The defect of Fano $3$-folds

Abstract

This paper studies the rank of the divisor class group of terminal Gorenstein Fano $3$-folds. If $Y$ is not $\mathbb {Q}$-factorial, there is a small modification of $Y$ with a second extremal ray; Cutkosky, following Mori, gave an explicit geometric description of contractions of extremal rays on terminal Gorenstein $3$-folds. I introduce the category of weak-star Fanos, which allows one to run the Minimal Model Program (MMP) in the category of Gorenstein weak Fano $3$-folds. If $Y$ does not contain a plane, the rank of its divisor class group can be bounded by running an MMP on a weak-star Fano small modification of $Y$. These methods yield more precise bounds on the rank of $\operatorname {Cl} Y$ depending on the Weil divisors lying on $Y$. I then study in detail quartic $3$-folds that contain a plane and give a general bound on the rank of the divisor class group of quartic $3$-folds. Finally, I indicate how to bound the rank of the divisor class group of higher genus terminal Gorenstein Fano $3$-folds with Picard rank $1$ that contain a plane.