Geography of log models: theory and applications

Abstract

An introduction to geography of log models with applications to positive cones of FT varieties and to geometry of minimal models and Mori fibrations.Both authors were partially supported by NSF grant DMS-0701465. 1 2 Fixed equivalence. Boundaries B, B ′ ∈ B S are fix equivalent and we write B ∼ fix B ′ if the fixed parts F (B), F (B ′ ) of both b-divisors K + B log , K + B ′log have the same signatures of their multiplicities, that is: for any prime b-divisor D,Note that all the relations above are defined on the whole B S .The resulting models obtained by the LMMP are at least projective, and even slt by the slt LMMP. An abstract definition of resulting models gives a larger class of them. However, according to the following it is enough to consider only the good ones: projective Q-factorial wlc models, e.g., slt wlc models.Proposition 2.1. The equivalence ∼ wlc for projective Q-factorial wlc models is the same as that for all wlc models.Proof. Any non-Q-factorial wlc model can be obtained by a crepant contraction of some Q-factorial wlc model that gives a wlc model by the semiampleness. Any Q-factorial nonprojective wlc model can be obtained from a projective and Q-factorial wlc model by a (generalized) log flop. Both constructions use the slt LMMP and [Sho4]. Thus if the pairs have the same projective and Q-factorial, numerically equivalent models, they have the same numerically equivalent wlc models.The converse statement means that each projective and Q-factorial wlc model of (X/Z, B) will be also projective and Q-factorial wlc model of (X/Z, B ′ ) when B ∼ wlc B ′ (cf. the proof of Lemma 2.7).Convention 2.2. The model equivalences ∼ mod , ∼ wlc are defined for projective and Q-factorial resulting models of pairs. In fact, we can even use slt wlc models since Proposition 2.1 holds for slt wlc models too. In most of constructions below, we use slt wlc resulting models. However, slt wlc models are not stable for limits of boundaries (see Lemma 2.9).The following relations hold for model equivalences.Proposition 2.3. We have the following implications:Proof. Immediate by definition and the semiampleness 7.1. 3In general, ∼ fix ⇒∼ mod ⇒∼ wlc ⇐∼ lcm . For example, ∼ fix ⇒∼ mod follows from the fact that the small modifications of flopping facets preserve the ∼ fix equivalence relation but not ∼ mod (see Theorem 5.9, 5.11).Define the subset of B S :Equivalently, we can also use the condition: ν(X/Z, B) ≥ 0 (see Numerical Kodaira dimension in Section 7). Moreover, by the semiampleness 7.1, this is equivalent to the nonnegativity of (invariant) Kodaira dimension (see Kodaira dimension in Section 7): κ(X/Z, B) ≥ 0.Proposition 2.3 implies that ∼ wlc is the finest of the model equivalences on N S . Thus, for the pairs with wlc models, the finiteness and polyhedral properties of all the equivalence relations follow from those of ∼ wlc . All the above equivalence relations, except for ∼ mod , are not interesting outside N S because they are trivial on B S \ N S . It is expected that many similar properties hold for ∼ mod...

The rationality problem for conic bundles

“…The three-dimensional version of this program was outlined in preprints [Sar87], [Sar89], [Rei91] and a complete proof (with termination) was given in [Cor95] (see also [IS05], [SC11]). In higher dimensions the program is established in a weaker form [HM13].…”

“…Termination of the Sarkisov program in [Cor95, Theorem 6.1] was proved by reduction to the log canonical case [Ale94a], [HMX14]. The approach of [HM13] is different (see also [IS05], [SC11]).…”

“…K Z ·C i ≤ 0 (thus Z is the central object of the corresponding link [SC11]). This inequality holds and can be checked directly in many cases.…”

“…For Q-del Pezzo fibrations there are partial results on construction of standard models [Cor96], [Kol97] and birational rigidity, see e.g. [Puk98], [Puk13,, [Che05], [Sob02], [Gri00], [SC11].…”

Duality of the cones of divisors and curves

“…It is known that the cone Amp k (X) for a d-dimensional FT variety X is rational polyhedral for k = 1, d − 1 and d [19]. More generally, we may ask the following.…”