Yuri Prokhorov
We study Mori fiber spaces over a two-dimensional base which satisfy the semistability assumption. As an application of our technique we give a new proof of the existence of semistable 3-fold flips.
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The semistable minimal model program is a special case of the minimal model program concerning 3-folds fibred over a curve and birational morphisms preserving this structure. We classify semistable divisorial contractions which contract the exceptional divisor to a normal point of a fibre. Our results can be applied to describe compact moduli spaces of surfaces.
We prove a relative version of the theorem of Cho, Miyaoka and Shepherd-Barron: a Mori fibre space of maximal length is birational to a projective bundle.
Given any field $k$ (not necessarily perfect), we study the smoothing of a semistable Fano variety over $k$. In characteristic 0, the reduced semistable Fano degenerate fibers of Mori fibrations are classified. In positive characteristic, under a suitable $W_2$ lifting assumption, we prove that a semistable Fano variety always appears as a degenerate fiber in a semistable family if it has a global log structure (in the sense of Fontaine-Illusie-Kato) of semistable type. A similar smoothing result over a mixed characteristic base is also obtained.
We study Mori's three-dimensional contractions $f\colon X\to Z$. It is proved that on the ``good'' model $(\bar X,\bar S)$ the exceptional divisor has no elliptic components of $Diff_{\bar S}$ with coefficients $\ge 6/7$.
It was conjectured by McKernan and Shokurov that for all Mori contractions from X to Y of given dimensions, for any positive epsilon there is a positive delta, such that if X is epsilon-log terminal, then Y is delta-log terminal. We prove this conjecture in the toric case and discuss the dependence of delta on epsilon, which seems mysterious.
Divisorial Mori contractions of submaximal length
本文改变极大长度假设，证明次极大长度的除子型Mori收缩的例外轨迹双有理等价于射影丛或二次曲面丛，拓展了Mori收缩相关刻画研究。
Let $X$ be a variety with at most terminal $\mathbb Q$-factorial singularities of dimension $n$. We study local contractions $f:X\to Z$ supported by a $\mathbb Q$-Cartier divisor of the type $K_X+ τL$, where $L$ is an $f$-ample Cartier divisor and $τ\geq 0$ is a rational number. Equivalently, $f$ is a Fano-Mori contraction associated to an extremal face in $\overline {NE(X)}_{K_X+τL = 0}$; these maps naturally arise in the context of the minimal model program. We prove that, if $τ> (n-3) >0$, the general element $X' \in |L|$ is a variety with at most terminal singularities. Then we apply this …
We treat equivariant completions of toric contraction morphisms as an application of the toric Mori theory. For this purpose, we generalize the toric Mori theory for non-$\mathbb Q$-factorial toric varieties. So, our theory seems to be quite different from Reid's original combinatorial toric Mori theory. We also explain various examples of non-$\mathbb Q$-factorial contractions, which imply that the $\mathbb Q$-factoriality plays an important role in the Minimal Model Program. Thus, this paper completes the foundations of the toric Mori theory and show us a new aspect of the Minimal Model Prog…
Let X be a projective variety with terminal singularities and let L be an ample Cartier divisor on X. We prove that if f is a birational contraction associated to an extremal ray $ R \subset \bar {NE(X)}$ such that R.(K_X+(n-2)L)<0, then f is a weighted blow-up of a smooth point. We then classify divisorial contractions associated to extremal rays R such that R.(K_X+rL)<0, where r is a non-negative rational number, and the fibres of f have dimension less or equal to r+1.
We will give a criterion to assure that an extremal contraction of a K3 surface which is not a Mori Dream Space produces a singular surface which is a Mori Dream Spaces. We list the possible Néron--Severi groups of K3 surfaces with this property and an extra geometric condition such that the Picard number is greater then or equal to 10. We give a detailed description of two geometric examples for which the Picard number of the K3 surface is 3, i.e. the minimal possible in order to have the required property. Moreover we observe that there are infinitely many examples of K3 surfaces with the re…
The main goal of this paper is to study varieties with the best possible Mori theoretic properties (measured by the existence of a certain decomposition of the cone of effective divisors). We call such a variety a Mori Dream Space. There turn out to be many examples, including quasi-smooth projective toric (or more generally, spherical) varieties, many GIT quotients, and log Fano 3-folds. We characterize Mori dream spaces as GIT quotients of affine varieties by a torus in a manner generalizing Cox's construction of toric varieties as quotients of affine space. Via the quotient description, the…
Let $f:X\to S$ be an extremal contraction from a threefolds with terminal singularities onto a surface (so called Mori conic bundle). We study some particular cases of such contractions: quotients of usual conic bundles and index two contractions. Assuming Reid's general elephants conjecture we also obtain a rough classification. We present many examples.
We give a method for constructing maps from a non-commutative scheme to a commutative projective curve. With the aid of Artin-Zhang's abstract Hilbert schemes, this is used to construct analogues of the extremal contraction of a $K$-negative curve with self-intersection zero on a smooth projective surface. This result will hopefully be useful in studying Artin's conjecture on the birational classification of non-commutative surfaces. As a non-trivial example of the theory developed, we look at non-commutative ruled surfaces and, more generally, at non-commutative $\PP^1$-bundles. We show in pa…
We prove the following results for projective klt pairs of dimension $3$ over an algebraically closed field of char $p>5$: the cone theorem, the base point free theorem, the contraction theorem, finiteness of minimal models, termination with scaling, existence of Mori fibre spaces, etc.
We study the local structure of Mori contractions $f\colon X\to Z$ of relative dimension one under an additional assumption that there exists a reduced divisor $S$ such that $K_X+S$ is plt and anti-ample.
We investigate the birational geometry (in the sense of Mori's program) of the moduli space of rank 2 semistable parabolic vector bundles on a rational curve. We compute the effective cone of the moduli space and show that all birational models obtained by Mori's program are also moduli spaces of parabolic vector bundles with certain parabolic weights.
We identify a set of initial rational contractions of fiber type on $\overline{M}_{0,6}$. Our proof uses a new algorithm we develop for verifying descriptions of the cone of effective divisors on varieties without elementary rational contractions of fiber type. On Mori dream spaces, our algorithm identifies faces of the cone of nef curves associated to initial rational contractions of fiber type.
We prove the existence of a Mori contraction on a compact Kaehler threefold whose canonical bundle is (analytically) not nef if the threefold can be approximated by projective threefolds or if the algebraic dimension is 2.
Small modifications of Mori dream spaces arising from ${\mathbb C}^*$-actions
本研究将带 ${\bb C}^*$-作用的射影簇小修正与GIT商关联，用Białynicki-Birula胞腔闭包处的flip构造双有理等变修正，完整刻画特定光滑簇的吹张修正结构，并给出齐性簇上的构造实例。
We investigate nef and movable cones of hypersurfaces in Mori dream spaces. The first result is: Let $Z$ be a smooth Mori dream space of dimension at least four whose extremal contractions are of fiber type of relative dimension at least two and let $X$ be a smooth ample divisor in $Z$, then $X$ is a Mori dream space as well. The second result is: Let $Z$ be a Fano manifold of dimension at least four whose extremal contractions are of fiber type and let $X$ be a smooth anti-canonical hypersurface in $Z$, which is a smooth Calabi--Yau variety, then the unique minimal model of $X$ up to isomorph…
We show that for $α\in (2/3, 7/10)$, the log canonical model $\bar M_4(α)$ of the pair $(\bar M_4, αδ)$ is isomorphic to the moduli space $\bar M_4^{hs}$ of h-semistable curves, and that there is a birational morphism $Ξ: \bar M_4^{hs} \to \bar M_4(2/3)$ that contracts the locus of curves $C_1\cup_p C_2$ consisting of genus two curves meeting in a node $p$ such that $p$ is a Weierstrass point of $C_1$ or $C_2$. To obtain this morphism, we construct a compact moduli space $\bar M_{2,1}^{hs}$ of pointed genus two curves that have nodes, ordinary cusps and tacnodes as singularity, and prove that …
Existence of moduli spaces for algebraic stacks
本文给出代数栈存在良好模空间的充要条件，证明带 $Θ$-分层的半稳定约化定理，推广 Keel--Mori 定理并应用于构造半稳定 $\mathcal{G}$-丛和阿贝尔范畴对象的紧合模空间。
In this paper, we study the divisor theory of the Simpson moduli space of semistable sheaves of dimension 1 on the projective plane. We prove that these spaces are all Mori dream spaces, and calculate their nef cones. We also study the effective cones of these spaces for most choices of numerical invariants. As a consequence, we work out precisely when two such spaces are isomorphic.
Local symplectic contractions are resolutions of singularities which admit symplectic forms. Four dimensional symplectic contractions are (relative) Mori Dream Spaces. In particular, any two such resolutions of a given singularity are connected by a sequence of Mukai flops. We discuss the cone of movable divisors on such a resolution; its faces are determined by curves whose loci are divisors, we call them essential curves. The movable cone is divided into nef chambers which are related to different resolutions; this subdivision is determined by classes of 1-cycles. We also study schemes param…
We study semistable extremal threefold neighborhoods 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 neighborhoods of type k1A. In fact we show that they belong to the same deformation family as k2A, and we explicitly construct the universal family of extremal neighborhoods. This construction follows very closely Mori's division algorithm, which we interpret as a sequence of mutations in the cluster algebra of rank 2 with general coefficients. We identify, in …
The Mori cone of certain Hassett spaces
本文证明，通用族为P^1丛的Hassett空间的Mori锥由一维分层生成，推广了Bolognesi和Massarenti建立的对称GIT商(P^1)^n//PGL_2的情形。
This paper is devoted to extend some Hu-Keel results on Mori dream spaces (MDS) beyond the projective setup. Namely, $\Q$-factorial algebraic varieties with finitely generated class group and Cox ring, here called \emph{weak} Mori dream spaces (wMDS), are considered. Conditions guaranteeing the existence of a neat embedding of a (completion of a) wMDS into a complete toric variety are studied, showing that, on the one hand, those which are complete and admitting low Picard number are always projective, hence Mori dream spaces in the sense of Hu-Keel. On the other hand, an example of a wMDS doe…
We study the birational geometry of a Fano 4-fold X from the point of view of Mori dream spaces; more precisely, we study rational contractions of X. Here a rational contraction is a rational map f: X-->Y, where Y is normal and projective, which factors as a finite sequence of flips, followed by a surjective morphism with connected fibers. Such f is called elementary if the difference of the Picard numbers of X and Y is 1. We first give a characterization of non-movable prime divisors in X, when X has Picard number at least 6; this is related to the study of birational and divisorial elementar…
We construct a sequence of modular compactifications of the space of marked trigonal curves by allowing the branch points to coincide to a given extent. Beginning with the standard admissible cover compactification, the sequence first proceeds through contractions of the boundary divisors and then through flips of the so-called Maroni strata, culminating in a Fano model for even genera and a Fano fibration for odd genera. While the sequence of divisorial contractions arises from a more general construction, the sequence of flips uses the particular geometry of triple covers. We explicitly desc…
Bridgeland stability conditions and skew lines on $\mathbb{P}^3$
本文研究$\mathbb{P}^3$上Hilbert概型中Bridgeland稳定性的穿墙现象，发现两面墙，证明每次穿墙对应光滑的$K$-负极端收缩，从而模空间是射影的。
Semiampleness on Jacobian elliptic surfaces of Kodaira dimension one
本文研究 Kodaira 维数为一的 Jacobian 椭圆曲面，在可约纤维条件下给出 Mori dream 曲面的判据，并构造反例说明奇数纤维时结论不成立。
Let $f\colon X\to Z$ be a Mori fibre space. McKernan conjectured that the singularities of $Z$ are bounded in terms of the singularities of $X$. Shokurov generalised this to pairs: let $(X,B)$ be a klt pair and $f\colon X\to Z$ a contraction such that $K_X+B\sim_\R 0/Z$ and that the general fibres of $f$ are Fano type varieties; adjunction for fibre spaces produces a discriminant divisor $B_Z$ and a moduli divisor $M_Z$ on $Z$. it is then conjectured that the singularities of $(Z,B_Z+M_Z)$ are bounded in terms of the singularities of $(X,B)$. We prove Shokurov conjecture when $(F,\Supp B_F)$ b…
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