Flipping surfaces

Abstract

We study semistable extremal 3-fold neighborhoods, which are fundamental building blocks of birational geometry, following earlier work of Mori, Kollár, and Prokhorov. We classify possible flips and extend Mori's algorithm for computing flips of extremal neighborhoods of type k2A to more general k1A neighborhoods. The novelty of our approach is to show that k1A belong to the same deformation family as k2A, in fact we explicitly construct the universal family of extremal neighborhoods. This construction follows very closely Mori's division algorithm, which can be interpreted as a sequence of mutations in the cluster algebra. We identify, in the versal deformation space of a cyclic quotient singularity, the locus of deformations such that the total space admits a (terminal) antiflip. We show that these deformations come from at most two irreducible components of the versal deformation space. As an application, we give an algorithm for computing stable one-parameter degenerations of smooth projective surfaces (under some conditions) and describe several components of the Kollár-Shepherd-Barron-Alexeev boundary of the moduli space of smooth canonically polarized surfaces of geometric genus zero. s 1 In some papers C is allowed to be reducible, however we can always reduce to the irreducible case by factoring f locally analytically over Q ∈ Y, see [K88, 8.4].2 This is a special case of the general elephant conjecture of Miles Reid. 2 6 Acknowledgements. . We would like to thank I. Dolgachev, J. Kollár, S. Mori, and M. Reid for helpful discussions. In particular, M. Reid explained to us in detail his joint work with G. Brown which describes explicitly the graded ring of a k2A flip [BR13].Contents m 2 1(1, m 1 a 1 − 1)) for some m 0 , m 2 ∈ N such that m 0 ≡ m 2 (m 1 a 1 − 1) mod m 2 1 . 9

Optimal bounds for T-singularities in stable surfaces

“…Most of them are rational, and nearly all are T-singularities of long length. By means of the explicit MMP in [HTU17], we can realize these rational examples W in such a way that S = P 2 ; for details see [Urz16b]. In many cases the curve π(C) has degree 7.…”

“…Thus the general fibre W has K W ample, and it has one T-singularity 1 dn 2 (1, dna − 1) of length r. Then we can bound r − d as in Theorem 1.1 since κ(S) ≤ κ(S ), where S is the minimal model of the minimal resolution of W . This can be proved by means of the stable MMP [HTU17], and the hierarchy of Kodaira dimensions in [K92,Lemma 2.4]. We remark that in any case the bound can be taken as 4K 2 W , but one can be precise after performing MMP.…”

Smoothly Embedded Rational Homology Balls

“…We review some basics of divisorial contractions and flips in minimal model program from Kollár-Mori [5] and HTU [3] Definition 3.1. A three dimensional extremal neighborhood is a proper birational morphism f : (C ⊂ W ) → (Q ∈ Z ) satisfying the following properties:…”

“…Then it is not difficult to show that (C ⊂ W ) → (Q ∈ Z ) is a flipping extremal neighborhood of type mk1A; cf. HTU [3].…”

Threefold extremal curve germs with one non-Gorenstein point

“…For example, in the exceptional flipping case, [KM92] provides relatively simple computations of flipped variety. For semistable germs these computations become more explicit; in the (k2A) case, from the general member H ∈ |O X | one can decide whether (X, C) is flipping or divisorial [Mor02, Corollary 4.1] and furthermore describe the flipped variety [Mor02,Theorem 4.7] and Z [Mor02, Theorem 4.5], respectively; the (k1A) case is similarly treated by [HTU17] under additional assumption "b 2 (X s ) = 1" (see 11.4.6). According to local classification (see Propositions 5.4 and 5.5), a semistable extremal curve germ of type (k1A) can be of type (IA ∨ ) or (IA).…”