We construct explicit examples of elementary extremal contractions, both birational and of fiber type, from smooth projective n-dimensional varieties, n\geq 4, onto smooth projective varieties, arising from classical projective geometry and defined over sufficiently small fields, not necessarily algebraically closed.
The examples considered come from particular special homaloidal and subhomaloidal linear systems, which usually are degenerations of general phenomena classically investigated by Bordiga, Severi, Todd, Room, Fano, Semple and Tyrrell and more recently by Ein and Shepherd-Barron.
The first series of examples is associated to particular codimension 2 determinantal smooth subvarieties of P^m, 3\leq m\leq 5. We get another series of examples by considering special cubic hypersurfaces through some surfaces in P^5, or some 3-folds in P^7 having one apparent double point. The last examples come from an intriguing birational elementary extremal contraction in dimension 6, studied by Semple and Tyrrell and fully described in the last section.
We consider extremal contractions on smooth Fano fourfolds whose second Chern character is positive. We show that such contractions can neither be of fiber type nor contract a divisor to a point.
We show that any extremal contraction from a smooth projective variety with dimension less than or equal to three appears as a moduli space of (semi)stable objects in the derived category of coherent sheaves.
We show that a toric Fano contraction associated to an extremal ray whose length is greater than the dimension of its fiber is a projective space bundle.
We prove that two Springer maps over a nilpotent orbit closure with the same degree are connected by stratified Mukai flops and the latter is obtained by extremal contractions of a natural resolution of the nilpotent orbit closure.
In this paper, we give explicit combinatorial descriptions for toric extremal contractions under the relative setting, where varieties do not need to be complete. Fujino's completion theorem is the key to the main result. As applications, we can generalize some of Mustata's results related to Fujita's conjecture on toric varieties for the relative case.
We discuss the problem of classifying birational extremal contractions of smooth threefolds where the canonical bundle is trivial along the curves contracted, in the case when a divisor is contracted to a point. We prove the analytic rigidity of the contraction in the case this divisor is normal of degree >= 5 (i.e. we show that the analytic structure of the contraction is completely determined by the isomorphism class of the exceptional locus and its normal bundle in X). This was previously known only in the case of smooth exceptional locus.
In this paper, we obtain a complete classification of smooth toric Fano varieties equipped with extremal contractions which contract divisors to curves for any dimension. As an application, we obtain a complete classification of smooth projective toric varieties which can be equivariantly blown-up to Fano along curves.
We survey recent results on ample cones and birational contractions of holomorphic symplectic varieties of K3 type, focusing on explicit constructions and concrete examples.
Let X be the blow-up of a smooth projective 4-fold Y along a smooth curve C and let E be the exceptional divisor. Assume that X is a Fano manifold and has an elementary extremal contraction $ϕ: X \to Z$ of (3,1)-type such that E is $ϕ$-ample (recall that a contraction map for a 4-fold is called (3,1)-type if the exceptional locus is a divisor and its image is a curve). We show that if the exceptional divisor of $ϕ$ is smooth, then Y is isomorphic to $\mathbb{P}^{4}$ and C is an elliptic curve of degree 4.
We consider $K_X$-negative extremal contractions $f\colon X\to (Z,o)$, where $X$ is an algebraic threefold with only $ε$-log terminal Q-factorial singularities and $(Z,o)$ is a two (resp., one)-dimensional germ. The main result is that $K_X$ is 1, 2, 3, 4 or 6-complementary or we have, so called, exceptional case and then the singularity $(Z\in o)$ is bounded (resp., the multiplicity of the central fiber $f^{-1}(o)$ is bounded).
A construction of smooth varieties admitting small contractions
Graduate School of Mathematical Sciences, the University of Tokyo（东京大学数学科学研究生院）；Department of Mathematics, School of Science and Engineering, Waseda University（早稻田大学理工学术院数学系）
Let $(X, C)$ be a germ of a threefold $X$ with terminal singularities along an irreducible reduced complete curve $C$ with a contraction $f: (X, C)\to (Z, o)$ such that $C=f^{-1}(o)_{red}$ and $-K_X$ is ample. Assume that $(X, C)$ contains a point of type (IIA) and that a general member $H\in |O_X|$ containing $C$ is normal. We classify such germs in terms of $H$.
Let $(X, C)$ be a germ of a threefold $X$ with terminal singularities along an irreducible reduced complete curve $C$ with a contraction $f: (X, C)\to (Z, o)$ such that $C=f^{-1}(o)_{red}$ and $-K_X$ is ample. Assume that $(X, C)$ contains a point of type (IIA). This paper continues our study of such germs containing a point of type (IIA) started in our previous paper arXiv:1601.07671.
Let $(X,C)$ be a germ of a threefold $X$ with terminal singularities along an irreducible reduced complete curve $C$ with a contraction $f: (X,C)\to (Z,o)$ such that $C=f^{-1}(o)_{red}$ and $-K_X$ is ample. Assume that $(X,C)$ contains a point of type (IC) or (IIB). We complete the classification of such germs in terms of a general member $H\in |\mathcal O_X|$ containing $C$.
This paper has been withdrawn by the author. It is not true that plt blow-up of toric singularity is toric (in dimension 3 too) and the first main theorem is incorrect also. The error is in deformation argument. I am grateful to Ivan Cheltsov for discussions.
The Castelnuovo-Mumford regularity of varieties of degree r and dimension n in the r-dimensional projective space that have an extremal secant line, is at least d-r+n+1. We classify these varieties and show that their regularity is exactly d-r+n+1, as predicted by the regularity conjecture.
Let $(X,C)$ be a germ of a threefold $X$ with terminal singularities along an irreducible reduced complete curve $C$ with a contraction $f: (X,C)\to (Z,o)$ such that $C=f^{-1}(o)_{\red}$ and $-K_X$ is ample. Assume that a general member $F\in |-K_X|$ meets $C$ only at one point $P$ and furthermore $(F,P)$ is Du Val of type A if index$(X,P)=4$. We classify all such germs in terms of a general member $H\in |O_X|$ containing $C$.
We propose a general framework governing the intersection properties of extremal rays of irreducible holomorphic symplectic manifolds under the Beauville-Bogomolov form. Our main thesis is that extremal rays associated to Lagrangian projective subspaces control the behavior of the cone of curves. We explore implications of this philosophy for examples like Hilbert schemes of points on K3 surfaces and generalized Kummer varieties. We also collect evidence supporting our conjectures in specific cases.
We classify extremal divisorial contraction which contracts a divisor to a curve from a smooth fourfold. We prove the exceptional divisor is $\Bbb P^2$bundle or quadric bundle over a smooth curve and the contraction is the blowing up along the curve.
The present paper provides a geometric characterization of complete flag varieties for semisimple algebraic groups. Namely, if $X$ is a Fano manifold whose all elementary contractions are $\mathbb P^1$-fibrations then $X$ is isomorphic to the complete flag manifold $G/B$ where $G$ is a semi-simple Lie algebraic group and $B$ is a Borel subgroup of $G$.
An extremal curve germ is the analytic germ of a threefold with terminal singularities along a reduced complete curve admitting a contraction whose fibers have dimension at most one. The aim of the present paper is to review the results concerning those contractions whose central fiber is irreducible and contains only one non-Gorenstein point.
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 consider birational projective contractions f:X -> Y from a smooth symplectic variety X over the complex numbers. We first show that exceptional rational curves on X deform in a family of dimension at least 2n-2. Then we show that these contractions are generically coisotropic, provided X is projective. Then we specialize to contractions with 1-dimensional exceptional fibres. We classify them in a natural way in terms of (Γ, G), where Γis a Dynkin diagram of type A_l, D_l or E_l and G is a permutation group of automorphisms of Γ. The 1-dimensional fibres do not degenerate, except if the con…
Let $f\colon X\to S$ be an extremal contraction from a threefold with only terminal singularities to a surface. We study local analytic structure such contractions near degenerate fiber $C$ in the case when $C$ is irreducible and $X$ has on $C$ only one non-Gorenstein point.
Relations among fundamental invariants play an important role in algebraic geometry. It is known that an $n$-dimensional variety of general type with nef canonical divisor and canonical singularities, whose image $Y$ under the canonical map is of maximal dimension, satisfies $K_X^n \ge 2 (p_g-n)$. We investigate the very interesting extremal situation $K_X^n=2(p_g-n)$, which appears in a number of geometric situations. Since these extremal varieties are natural higher dimensional analogues of Horikawa surfaces, we name them Horikawa varieties. These varieties have been previously dealt with in…
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…
We discuss the flatness property of some fiber type contractions of complex smooth projective varieties of arbitrary dimensions. We relate the flatness of some morphisms having one-dimensional fibers with their conic bundles structures, also in the general case in which some mild singularities of the varieties are admitted.
Let $g : X \to Y$ be the contraction of an extremal ray of a smooth projective 4-fold $X$ such that $\dim Y=3$. Then $g$ may have a finite number of 2-dimensional fibers. We shall classify those fibers. Especially we shall prove that any two points of such a fiber is joined by a chain of rational curves of length at most 2 with respect to $-K_X$, and that $|-K_X|$ is $g\text{-free}$.
In this paper we discuss some recent results about extremal contractions of complex algebraic varieties. These are proper surjective maps, $ϕ: X\longrightarrow Z$, of normal varieties with connected fibers such that $X$ has mild singularities (in particular $X$ is $Q$-Gorenstein) and the anticanonical divisor of $X$ is $ϕ$ ample. It will appear on the Proceedings of AMS-SRI on Algebraic Geometry, Santa Cruz 1995.
From Smyth's classification, modular compactifications of pointed smooth rational curves are indexed by combinatorial data, so-called extremal assignments. We explore their combinatorial structures and show that any extremal assignment is a finite union of atomic extremal assignments. We discuss a connection with the birational geometry of the moduli space of stable pointed curves. As applications, we study three special classes of extremal assignments: smooth, toric, and invariant with respect to the symmetric group action. We identify them with three combinatorial objects: simple intersectin…
In this paper we study a local structure of extrmal contractions $f\colon X\to S$ from threffolds $X$ with only terminal singularities onto a surface $S$. If the surface $S$ is non-singular and $X$ has a unique non-Gorenstein point on a fiber we prove that either the linear system $|-K_X|$, $|-2K_X|$ or $|-3K_X|$ contains a "good" divisor.
We classify smooth Fano manifolds X with the Picard number $ρ_X \geq 3$ such that there exists an extremal ray which has a birational contraction that maps a divisor to a point.
Let X be a Fano variety of dimension n, pseudoindex i_X and Picard number ρ_X. A generalization of a conjecture of Mukai says that ρ_X(i_X-1)\le n. We prove that the conjecture holds if: a) X has pseudoindex i_X \ge \frac{n+3}{3} and either has a fiber type extremal contraction or does not have small extremal contractions b) X has dimension five.
In view of Mori theory, rational homogenous manifolds satisfy a recursive condition: every elementary contraction is a rational homogeneous fibration and the image of any elementary contraction also satisfies the same property. In this paper, we show that a smooth Fano $n$-fold with the same condition and Picard number greater than $n-6$ is either a rational homogeneous manifold or the product of $n-7$ copies of $\mathbb{P}^1$ and a Fano $7$-fold $X_0$ constructed by G. Ottaviani. We also clarify that $X_0$ has non-nef tangent bundle and in particular is not rational homogeneous.
In the framework of the problem of characterizing complete flag manifolds by their contractions, the complete flags of type $F_4$ and $G_2$ satisfy the property that any possible tower of Bott-Samelson varieties dominating them birationally deforms in a nontrivial moduli. In this paper we illustrate the fact that, at least in some cases, these deformations can be explained in terms of automorphisms of Schubert varieties, providing variations of certain isotropic structures on them. As a corollary, we provide a unified and completely algebraic proof of the characterization of complete flag mani…
We give the first examples of flat fiber type contractions of Fano manifolds onto varieties that are not weak Fano, and we prove that these morphisms are Fano conic bundles. We also review some known results about the interaction between the positivity properties of anticanonical divisors of varieties of contractions.
To a graph, Hausel and Proudfoot associate two complex manifolds, B and D, which behave, respectively like moduli of local systems on a Riemann surface, and moduli of Higgs bundles. For instance, B is a moduli space of microlocal sheaves, which generalize local systems, and D carries the structure of a complex integrable system. We show the Euler characteristics of these varieties count spanning subtrees of the graph, and the point-count over a finite field for B is a generating polynomial for spanning subgraphs. This polynomial satisfies a deletion-contraction relation, which we lift to a del…
Let $\f: X \ra Z$ be a proper surjective map from a smooth complex manifold $X$ onto a normal variety $Z$. If $\f$ has connected fibers and $-K_X$ is $\f$-ample then $\f$ is called a good contraction. In the present paper we study good contractions, fibers of which have dimension less or equal than two: after describing possible two dimensional isolated fibers we discuss their scheme theoretic structure and the geometry of $\f:X\ra Z$ nearby such a fiber. If $dimX=4$ and $\f$ is birational with an isolated 2 dimensional fiber then we obtain a complete description of $\f$. We provide also a des…
We develop a general theory for the existence of extremal Kähler metrics of Poincaré type in the sense of Auvray, defined on the complement of a toric divisor of a polarized toric variety. In the case when the divisor is smooth, we obtain a list of necessary conditions which must be satisfied for such a metric to exist. Using the explicit methods of Apostolov-Calderbank-Gauduchon together with the computational approach of Sektnan, we show that on a Hirzebruch complex surface the necessary conditions are also sufficient. In particular, on such a complex surface the complement of the infinity s…
Let C be a smooth curve on an index 1 terminal 3-fold. We investigate the existence of extremal terminal divisorial contractions Y-->X that contract an irreducible surface E to C. We consider cases in respect to the singularities of the general hypersurface section S of X through C. We completely classify such contractions in the case that S has A_i, i < 4 and D_2n for any n, type singularities.
For any $n\geq 3$, we prove that there exist equivalences between these apparently unrelated objects: irreducible $n$-dimensional non degenerate projective varieties $X\subset \mathbb P^{2n+1}$ different from rational normal scrolls and 3-covered by twisted cubic curves, up to projective equivalence; quadro-quadric Cremona transformations of $ \mathbb P^{n-1}$, up to linear equivalence; $n$-dimensional complex Jordan algebras of rank three, up to isotopy. We also provide some applications to the classification of particular classes of varieties in the class defined above and of quadro-quadric …
As proved recently in [PT], for varieties $X^{r+1}\subset \mathbb P^N$ such that through $n\geq 2$ general points there passes an irreducible curve $C$ of degree $δ\geq n-1$ we have $N\leq π(r,n,δ+r(n-1)+2)$, where $π(r,n,d)$ is the Castelnuovo-Harris bound function for the geometric genus of an irreducible non-degenerate variety $Y^r\subset\mathbb P^{n+r-1}$ of degree $d$. A lot of examples of varieties as in the title and attaining the previous bound for the embedding dimension are constructed from Castelnuovo varieties and were thus dubbed {\it of Castelnuovo type} in [PT], where it is also…
In this paper we study the geometry of mildly singular Fano varieties on which there is an effective prime divisor of Picard number one. Afterwards, we address the case of toric varieties. Finally, we treat the lifting of extremal contractions to universal covering spaces in codimension 1.
We study the birational geometry of $\bar{M}_{3,1}$ and $\bar{M}_{4,1}$. In particular, we pose a pointed analogue of the Slope Conjecture and prove it in these low-genus cases. Using variation of GIT, we construct birational contractions of these spaces in which certain divisors of interest -- the pointed Brill-Noether divisors -- are contracted. As a consequence, we see that these pointed Brill-Noether divisors generate extremal rays of the effective cones for these spaces.
All varieties, extremal contractions, singularities are divided on exceptional and non-exceptional ones. Roughly speaking, there are the infinite families of non-exceptional varieties, extremal contractions or singularities and only the finite number of the types of exceptional ones. This subdivision is well demonstrated by the example of Du Val singularities. There are two infinite series of non-exceptional singularities: $A_n$ and $D_n$ and only three types of exceptional ones: $E_6$, $E_7$ and $E_8$. Also the importance of exceptionality phenomenon follows from the next observation: A). If …
We give new estimates of lengths of extremal rays of birational type for toric varieties. We can see that our new estimates are the best by constructing some examples explicitly. As applications, we discuss the nefness and pseudo-effectivity of adjoint bundles of projective toric varieties. We also treat some generalizations of Fujita's freeness and very ampleness for toric varieties.
We use the correspondence between extremal Sasaki structures and weighted extremal Kahler metrics defined on a regular quotient of a Sasaki manifold, established by the first two authors, and Lahdili's theory of weighted K-stability in order to define a suitable notion of (relative) weighted K-stability for compact Sasaki manifolds of regular type. We show that the (relative) weighted K-stability with respect to a maximal torus is a necessary condition for the existence of a (possibly irregular) extremal Sasaki metric. We also compare weighted K-stability to the K-stability of the correspondin…
In this note we propose to show that the Kähler-Ricci flow fits naturally within the context of the Minimal Model Program for projective varieties. In particular we show that the flow detects, in finite time, the contraction theorem of any extremal ray and we analyze the singularities of the metric in the case of divisorial contractions for varieties of general type. In case one has a smooth minimal model of general type (i.e., the canonical bundle is nef and big), we show infinite time existence and analyze the singularities.
Starting from $\mathbb{C}^*$-actions on complex projective varieties, we construct and investigate birational maps among the corresponding extremal fixed point components. We study the case in which such birational maps are locally described by toric flips, either of Atiyah type or so called non-equalized. We relate this notion of toric flip with the property of the action being non-equalized. Moreover, we find explicit examples of rational homogeneous varieties admitting a $\mathbb{C}^*$-action whose weighted blow-up at the extremal fixed point components gives a birational map among two proj…
