Anticanonical Volumes of Q-Fano Threefolds

In dimension three, by Kollar-Miyaoka-Mori-Tagaki the set of Fano threefolds with at worst canonical singularities is bounded. In particular, there is an upper bound of the anticanonical volume -KX^3 of canonical Fano threefolds. By the early work on the classification of smooth Fano threefolds (Mori-Miyaoka) and Gorenstein canonical Fano threefolds (Prokhorov), it is known that -KX^3 is optimally no large than 72. However, the problem becomes much more difficult if X is not Gorenstein, except when Picard number of X is one where -KX^3 is less or equal to 125/2 (Prokhorov). In this proposal, we aim to find the optimal upper bound for anticanonical volume for canonical Fano threefolds. It is known from the minimal model program (BCHM) that we can equivalently study the anticanonical volume on weak Q-Fano threefolds, i.e., terminal Q-Gorenstein weak Fano varieties with at worst terminal singularities. By cone theorem, there is a KX-negative extremal contraction of X. If this is a del Pezzo fibration, then Picard rank of X is two and we have obtained that -KX^3 is no more than 72 by exploiting the geometry of extremal rays of X. However, if this contraction is a conic bundle, then Picard number can be bigger than two and it is not easy to reveal the interaction of geometry of different extremal contractions. Instead, we aim to study the geometry of conic bundle through its discriminant divisor. Let X->S be a conic bundle from a weak Q-Fano threefold. If X is Gorenstein, then S is smooth and this morphism is flat. In particular, either X->S is a P^1 bundle and there is a birational model X-->X' over S->S', where S' is a relative minimal model of S, so that X' remains weak Q-Fano and -KX'^3 is bigger than -KX^3. Or X->S has a non-rational discriminant divisor on S, where in this scenario one can estimate -KX^3 on S via the chern classes of the associated vector bundle. This approach is completed by Prokhorov, but does not generalize directly to non-Gorenstein case. There are some new inputs. It is established by Mori-Prokhorov that S as the base of a threefold conic bundle has at most type A singularities. Based on this, we can show that S is weak del Pezzo and there is a well-defined discriminant divisor. On the other hand, J. A. Chen has constructed a feasible Gorenstein resolution for threefold terminal singularities. Hence we can try to construct a Gorenstein birational model of X with explicitly described morphisms (Kawamata blowups) and see how the conic bundle can be transformed to a better behaved one. In this way, we aim to generalize the construction of Prokhorov for Gorenstein Fano threefolds to non-Gorenstein cases. The speculated results would include an interesting algorithm to modify a Q-Fano conic bundle and as a consequence the optimal upper bound of anticanonical volume for Q-Fano threefolds with a fibration structure. We expect this newly construction associated to Q-Fano bundles can also lead to other new results in three dimensional geometry.