An effective upper bound for anti-canonical volumes of canonical $\mathbb{Q}$-Fano threefolds

Abstract

We show that the anti-canonical volume of a canonical weak Q-Fano 3-fold is at most 324.

On the anti-canonical geometry of weak $\mathbb{Q}$-Fano threefolds, III

“…For simplicity, we will alternatively write a basket as a set of pairs with weights, say for example, (1,3), (3,7), (5,11)}.…”

“…[6, Lemma 4.2, Proposition 5.2]) but one has to replace X with another birational model. In this paper, by technique developed recently in [11], we give a new criterion as the following theorem. for some positive integer m, then | − mK X | is not composed with a pencil.…”

“…A structure theorem of terminal weak Q-Fano 3-folds. We recall the following structure theorem of terminal weak Q-Fano 3-folds from [11]. It plays the role of Fano-Mori triples as in [6].…”

“…(3) for any sufficiently large and divisible positive integer Here we remark that in the proof of [11,Proposition 4.1], Y is obtained by running a K-MMP on a Q-factorialization of X, so the induced map X Y is a contraction, that is, it does not extract any divisor.…”

“…Motivated by the classification theory of 3-dimensional algebraic varieties, we are interested in the study of explicit geometry of (weak) Q-Fano varieties with terminal or canonical singularities. In this direction there are a lot of works in the literature, see for example [15,16,3,17,4,18,5,6,11,10].…”