Yuri G. Prokhorov
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.
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We complete Mori's program for Kontsevich's moduli space of degree 2 stable maps to Grassmannian of lines. We describe all birational models in terms of moduli spaces (of curves and sheaves), incidence varieties, and Kirwan's partial desingularization.
We study certain pencils of del Pezzo surfaces generated by a smooth del Pezzo surface S of degree less or equal to 3 anti-canonically embedded into a weighted projective space P and an appropriate multiple of a hyperplane H. Our main observation is that every minimal model program relative to the morphism lifting such pencil on a suitable resolution of its indeterminacies preserves the open subset P H â A^3. As an application, we obtain projective completions of A^3 into del Pezzo fibrations over P^1 of every degree less or equal to 4. We also obtain completions of A^3 into Mori conic bundles…
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.
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.
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.
We obtain a sufficient condition for a Fano threefold with terminal singularities to have a conic bundle structure.
We study the geography and birational geometry of 3-fold conic bundles over P^2 and cubic del Pezzo fibrations over P^1. We discuss many explicit examples and raise several open questions. This paper was submitted to the proceedings of the "Fano conference" held in Torino in October 2002.
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 …
A $\mathbb Q$-conic bundle germ is a proper morphism from a threefold with only terminal singularities to the germ $(Z \ni o)$ of a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. We obtain the complete classification of $\mathbb Q$-conic bundle germs when the base surface germ is singular. This is a generalization of our previous paper math/0603736, which further assumed that the fiber over $o$ is irreducible.
A $\mathbb Q$-conic bundle is a proper morphism from a threefold with only terminal singularities to a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. We study the structure of $\mathbb Q$-conic bundles near their singular fibers. One corollary to our main results is that the base surface of every $\mathbb Q$-conic bundle has only Du Val singularities of type A (a positive solution of a conjecture by Iskovskikh). We obtain the complete classification of $\mathbb Q$-conic bundles under the additional assumption that the singular fiber is irreduc…
Birational transformations of threefold $\mathbf{Q}$-conic bundles
本文针对三维$\mathbf{Q}$-圆锥丛，提出一种算法将其变换为标准形式，该丛满足终端$\mathbf{Q}$-阶乘奇点、反典范除子$f$-丰沛及相对皮卡数1等条件。
Birational complexity and conic fibrations
本文研究了log Calabi-Yau对的极值复杂度与圆锥纤维化的关系，并引入了新的复杂度度量标准，探讨了其在birational超刚性、Fano超曲面等领域的应用。
We show that being a general fibre of a Mori fibre space is a rather restrictive condition for a Fano variety. More specifically, we obtain two criteria (one sufficient and one necessary) for a Q-factorial Fano variety with terminal singularities to be realised as a fibre of a Mori fibre space, which turn into a characterisation in the rigid case. We apply our criteria to figure out this property up to dimension three and on rational homogeneous spaces. The smooth toric case is studied and an interesting connection with K-semistability is also investigated.
Following Shokurov's ideas, we give a short proof of the following klt version of his result: termination of terminal log flips in dimension d implies that any klt pair of dimension d has a log minimal model or a Mori fibre space. Thus, in particular, any klt pair of dimension 4 has a log minimal model or a Mori fibre space.
We prove that a Mori dream space over a field of characteristic zero is of Calabi-Yau type if and only if its Cox ring has at worst log canonical singularities. By slightly modifying the arguments we also reprove the characterization of the varieties of Fano type by the log terminality of the Cox rings.
We study a particular kind of fiber type contractions between complex, projective, smooth varieties f:X->Y, called Fano conic bundles. This means that X is a Fano variety, and every fiber of f is isomorphic to a plane conic. Denoting by rho_{X} the Picard number of X, we investigate such contractions when rho_{X}-rho_{Y} is greater than 1, called non-elementary. We prove that rho_{X}-rho_{Y} is at most 8, and we deduce new geometric information about our varieties, depending on rho_{X}-rho_{Y}. Moreover, when X is locally factorial with canonical singularities and with at most finitely many no…
Spinor modifications of conic bundles and derived categories of 1-nodal Fano threefolds
本文针对平坦二次曲线丛定义旋量修改构造新二次曲线丛，证明其偶克利福德代数Morita等价且导出范畴中基底导出范畴的正交补等价，并以非因子1-结点素法诺三维簇为例展示该技术，实现奇点范畴吸收。
A Q-conic bundle germ is a proper morphism from a threefold with only terminal singularities to the germ $(Z \ni o)$ of a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. Building upon our previous paper [math/0603736], we prove the existence of a Du Val anti-canonical member under the assumption that the central fiber is irreducible.
We describe the set of Mori structures for a Fano 3-fold of index 2 and degree 1 (the double cone over the Veronese surface). In partiular, it is proved that such a Fano variety is not rational, the group of birational automorphisms coincides with the group of biregular automorphisms, and there are no structures of conic bundle.
To construct a resulting model in LMMP is sufficient to prove existence of log flips and their termination for certain sequences. We prove that LMMP in dimension $d-1$ and termination of terminal log flips in dimension $d$ imply, for any log pair of dimension $d$, the existence of a {\em resulting} log model: a strictly log minimal model or a strictly log terminal Mori log fibration, and imply existence of log flips in dimension $d+1$. As consequence, we prove existence of a resulting model of 4-fold log pairs, existence of log flips in dimension 5, and some fragments of Geography of log model…
We prove an analogue of Fujino and Mori's ``bounding the denominators'' in the log canonical bundle formula (see also Prokhorov and Shokurov) for Kawamata log terminal pairs of relative dimension one. As an application we prove that for a klt pair $(X,Δ)$ of Kodaira codimension one and dimension at most three such that the coefficients of $Δ$ are in a DCC set $\mathcal{A}$, there is a natural number $N$ that depends only on $\mathcal{A}$ for which the round down of $\N(K_X+Δ)$ induces the Iitaka fibration. We also prove a birational boundedness result for klt surfaces of general type.
In our preprint: "Algebraic surfaces with log-terminal singularities and nef anticanonical class and reflection groups in Lobachevsky spaces", Preprint Max-Planck-Institut für Mathematik, Bonn, (1989) MPI/89-28 (Russian), we had extended the diagram method to algebraic surfaces with nef anticanonical class (before, it was applied to Del Pezzo surfaces). The main problem here is that Mori polyhedron may not be finite polyhedral in this case. In the §4 of this preprint we had shown that this method does work for algebraic surfaces with nef anticanonical class and log terminal singularities. Rece…
MMP for Generalized Pairs on Kähler 3-folds
本文研究Kähler 3折的广义对MMP，证明了极值定理、翻转存在性、log终端模型等核心结果。
An empty simplex is a lattice simplex with only its vertices as lattice points. Their classification in dimension three was completed by White in 1964. In dimension four, the same task was started in 1988 by Mori, Morrison, and Morrison, with their motivation coming from the close relationship between empty simplices and terminal quotient singularities. They conjectured a classification of empty simplices of prime volume, modulo finitely many exceptions. Their conjecture was proved by Sankaran (1990) with a simplified proof by Bober (2009). The same classification was claimed by Barile et al. …
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