BOUNDING VOLUMES OF SINGULAR FANO THREEFOLDS

Abstract

Let$(X,\unicode[STIX]{x1D6E5})$be an$n$-dimensional$\unicode[STIX]{x1D716}$-klt log$\mathbb{Q}$-Fano pair. We give an upper bound for the volume$\text{Vol}(X,\unicode[STIX]{x1D6E5})=(-(K_{X}+\unicode[STIX]{x1D6E5}))^{n}$when$n=2$, or$n=3$and$X$is$\mathbb{Q}$-factorial of$\unicode[STIX]{x1D70C}(X)=1$. This bound is essentially sharp for$n=2$. The main idea is to analyze the covering families of tigers constructed in J. McKernan (Boundedness of log terminal fano pairs of bounded index, preprint, 2002,arXiv:0205214). Existence of an upper bound for volumes is related to the Borisov–Alexeev–Borisov Conjecture, which asserts boundedness of the set of$\unicode[STIX]{x1D716}$-klt log$\mathbb{Q}$-Fano varieties of a given dimension$n$.