A characterization of double covers of curves in terms of the ample cone of second symmetric product

Compact subvarieties with ample normal bundles, algebraicity and cones of cycles

In this note we study two features of submanifolds (subvarieties) with ample normal bundles in a compact Kähler manifold X. First, we study how algebraic X can be, i.e. we investigate the algebraic dimension. Second, we study curves with ample normal bundles in case X is projective. We prove in various cases that the class of the curve is in the interior of the Mori cone.

On the ample cone of a rational surface with an anticanonical cycle

Let $Y$ be a smooth rational surface and let $D$ be a cycle of rational curves on $Y$ which is an anticanonical divisor, i.e. an element of $|-K_Y|$. Looijenga studied the geometry of such surfaces $Y$ in case $D$ has at most five components and identified a geometrically significant subset $R$ of the divisor classes of square -2 orthogonal to the components of $D$. Motivated by recent work of Gross, Hacking, and Keel on the global Torelli theorem for pairs $(Y,D)$, we attempt to generalize some of Looijenga's results in case $D$ has more than five components. In particular, given an integral …

On nef subvarieties

The goal of this work is to study positivity of subvarieties with nef normal bundle in the sense of intersection theory. After Ottem's work on ample subschemes, we introduce the notion of a nef subscheme, which generalizes the notion of a subvariety with nef normal bundle. We show that restriction of a pseudoeffective (resp. big) divisor to a nef subvariety is pseudoeffective (resp. big). We also show that ampleness and nefness are transitive properties. We define the weakly movable cone as the cone generated by the pushforward of cycle classes of nef subvarieties via proper surjective maps. T…

膨胀边界与K3表面 ample锥的关联

Blown-up boundaries associated with ample cones of K3 surfaces

本文通过Bestvina-Mess公式研究K3表面自同构群的虚拟上同调维数，揭示了其与ample锥关联的膨胀边界覆盖维度之间的关系。

ample锥上的F-签名函数

The F-signature Function on the Ample Cone

本文在正特征代数闭域上，对全局F-正则射影簇X的ample锥定义了F-签名函数，证明其局部Lipschitz连续且可延拓至边界，并与体积函数建立有效比较，进而得出nef但不big除子上延拓为零的结论。

$\mathbb{P}^3$ 中超曲面上点的 Hilbert 概形的丰富锥

Ample cones of Hilbert schemes of points on hypersurfaces in $\mathbb{P}^3$

本文研究 $\mathbb{P}^3$ 中一般高次曲面上点的 Hilbert 概形，计算小 $n$ 时的丰富锥及含线情形的 nef 锥切片，并揭示五次情形中高重超平面截面的作用。

Seshadri constants on symmetric products of curves

Let X_g=C^{(2)}_g be the second symmetric product of a very general curve of genus g. We reduce the problem of describing the ample cone on X_g to a problem involving the Seshadri constant of a point on X_{g-1}. Using this we recover a result of Ciliberto-Kouvidakis that reduces finding the ample cone of X_g to the Nagata conjecture when g\ge 9. We also give new bounds on the the ample cone of X_g when g=5.

Towards the ample cone of $\mgn$

In this paper we study the ample cone of the moduli space $\mgn$ of stable $n$-pointed curves of genus $g$. Our motivating conjecture is that a divisor on $\mgn$ is ample iff it has positive intersection with all 1-dimensional strata (the components of the locus of curves with at least $3g+n-2$ nodes). This translates into a simple conjectural description of the cone by linear inequalities, and, as all the 1-strata are rational, includes the conjecture that the Mori cone is polyhedral and generated by rational curves. Our main result is that the conjecture holds iff it holds for $g=0$. More pr…

GIT 栈商的 Kleiman 准则

A Kleiman criterion for GIT stack quotients

本文将经典射影概形的Kleiman准则推广到带约化群作用的代数簇，用G-丰沛锥的GIT形变分解胞腔和到商栈[X/G]的拟映射分别替代原丰沛锥与曲线，得到了对应的丰沛性判定准则。

Numerical dimension and locally ample curves

In the paper \cite{Lau16}, it was shown that the restriction of a pseudoeffective divisor $D$ to a subvariety $Y$ with nef normal bundle is pseudoeffective. Assuming the normal bundle is ample and that $D|_Y$ is not big, we prove that the numerical dimension of $D$ is bounded above by that of its restriction, i.e. $κ_σ(D)\leq κ_σ(D|_Y)$. The main motivation is to study the cycle classes of "positive" curves: we show that the cycle class of a curve with ample normal bundle lies in the interior of the cone of curves, and the cycle class of an ample curve lies in the interior of the cone of movab…

$b$-除子类的正锥

Positive cones of $b$-divisor classes

本文利用 Seshadri 常数定义丰富 Cartier $b$-除子类，证明其构成 nef 锥内的凸锥，并刻画大除子类的凸锥及拉回性质。

Ample vector bundles and branched coverings

Given a covering f: X \to Y of projective manifolds, we consider the vector bundle E on Y given as the dual of f_*(Ø_X) / Ø_Y. This vector bundles often has positivity properties, e.g. E is ample when Y is projective space by a theorem of Lazarsfeld. In general however E will not be ample due to the geometry of Y. We prove various results when E is spanned, nef or generically nef, under some assumptions on the base Y.

On the $S_n$-invariant F-conjecture

By using classical invariant theory, we reduce the $S_{n}$-invariant F-conjecture to a feasibility problem in polyhedral geometry. We show by computer that for $n \le 19$, every integral $S_{n}$-invariant F-nef divisor on the moduli space of genus zero stable pointed curves is semi-ample, over arbitrary characteristic. Furthermore, for $n \le 16$, we show that for every integral $S_{n}$-invariant nef (resp. ample) divisor $D$ on the moduli space, $2D$ is base-point-free (resp. very ample). As applications, we obtain the nef cone of the moduli space of stable curves without marked points, and t…

Hyperbolic geometry of the ample cone of a hyperkahler manifold

Let $M$ be a compact hyperkahler manifold with maximal holonomy (IHS). The group $H^2(M, R)$ is equipped with a quadratic form of signature $(3, b_2-3)$, called Bogomolov-Beauville-Fujiki (BBF) form. This form restricted to the rational Hodge lattice $H^{1,1}(M,Q)$, has signature $(1,k)$. This gives a hyperbolic Riemannian metric on the projectivisation of the positive cone in $H^{1,1}(M,Q)$, denoted by $H$. Torelli theorem implies that the Hodge monodromy group $Γ$ acts on $H$ with finite covolume, giving a hyperbolic orbifold $X=H/Γ$. We show that there are finitely many geodesic hypersurfac…

Nef cones of Hilbert schemes of points on surfaces

Let X be a smooth projective surface of irregularity 0. The Hilbert scheme of n points on X parameterizes zero-dimensional subschemes of X of length n. In this paper, we discuss general methods for studying the cone of ample divisors on the Hilbert scheme. We then use these techniques to compute the cone of ample divisors on the Hilbert scheme for several surfaces where the cone was previously unknown. Our examples include families of surfaces of general type and del Pezzo surfaces of degree 1. The methods rely on Bridgeland stability and the Positivity Lemma of Bayer and Macri.

Okounkov bodies for ample line bundles with applications to multiplicities for group representations

Let $\mathscr{L} \rightarrow X$ be an ample line bundle over a complex normal projective variety $X$. We construct a flag $X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n=X$ of subvarieties for which the associated Okounkov body for $\mathscr{L}$ is a rational polytope. In the case when $X$ is a homogeneous surface, and the pseudoeffective cone of $X$ is rational polyhedral, we also show that the global Okounkov body is a rational polyhedral cone if the flag of subvarieties is suitably chosen. Finally, we provide an application to the asymptotic study of group representations.

On the cone of curves of an abelian variety

Let $X$ be a smooth projective variety over the complex numbers. One knows by the Cone Theorem that the closed cone of curves of $X$ is rational polyhedral whenever $c_1(X)$ is ample. For varieties $X$ such that $c_1(X)$ is not ample, however, it is in general difficult to determine the structure of $\bar NE(X)$. The purpose of this paper is to study the cone of curves of abelian varieties. Specifically, the abelian varieties $X$ are determined such that the closed cone $\bar NE(X)$ is rational polyhedral. The result can also be formulated in terms of the nef cone of $X$ or in terms of the sem…

K3^[n]或广义Kummer变形类型全纯辛变体的Kawamata–Morrison锥猜想证明

A proof of the Kawamata-Morrison Cone Conjecture for holomorphic symplectic varieties of K3^[n] or generalized Kummer deformation type

本文针对K3^[n]型及广义Kummer变形型的射影不可约全纯辛流形，证明了Kawamata–Morrison丰锥猜想的相应版本。

Balanced line bundles on Fano varieties

A conjecture of Batyrev and Manin relates arithmetic properties of varieties with ample anticanonical class to geometric invariants; in particular, counting functions defined by metrized ample line bundles and the corresponding asymptotics of rational points of bounded height are interpreted in terms of cones of effective divisors and certain thresholds with respect to these cones. This framework leads to the notion of balanced line bundles, whose counting functions, conjecturally, capture generic distributions of rational points. We investigate balanced line bundles in the context of the Mini…

ample锥的圆度与超凯勒流形上双拉格朗日纤维化的存在性

Roundness of the ample cone and existence of double Lagrangian fibrations on hyperkahler manifolds

本文证明任何满足 $b_2 > 4$ 的最大全纯超凯勒流形存在具有圆凯勒锥和符号 $(1,1)$ 的 Picard 格的形变，由此推出所有已知超凯勒流形具有双横截拉格朗日纤维化的形变，且 Kobayashi 度量在 Picard 秩非最大时消失。

The GIT-equivalence for $G$-line bundles

Let $X$ be a projective variety with an action of a reductive group $G$. Each ample $G$-line bundle $L$ on $X$ defines an open subset $X^{\rm ss}(L)$ of semi-stable points. Following Dolgachev and Hu, define a GIT-class as the set of algebraic equivalence classes of $L'$s with fixed $X^{\rm ss}(L)$. We show that the GIT-classes are the relative interiors of rational polyhedral convex cones, which form a fan in the $G$-ample cone. We also study the corresponding variations of quotients $X^{\rm ss}(L)//G$. This sharpens results of Thaddeus and Dolgachev-Hu.

The nef cone of toroidal compactifications of ${\cal A}_4$

We determine the cone of nef divisors on the Igusa and Voronoi compactifications of the moduli space of principally polarised abelian 4-folds. We also show that the canonical bundle on the Igusa compactification of A_4(n) is ample for n at least 3. Analogous results for A_2 and A_3 were earlier obtained by the first author but the case of A_4 is far less simple.

Automorphisms and moduli spaces of varieties with ample canonical class via deformations of abelian covers

By a recent result of Viehweg, projective manifolds with ample canonical class have a coarse moduli space, which is a union of quasiprojective varieties. In this paper, we prove that there are manifolds with ample canonical class that lie on arbitrarily many irreducible components of the moduli; moreover, for any finite abelian group $G$ there exist infinitely many components $M$ of the moduli of varieties with ample canonical class such that the generic automorphism group $G_M$ is equal to $G$. In order to construct the examples, we use abelian covers, i.e. Galois cover whose Galois group is …

Apollonian地毯与超Kähler流形的Kähler锥边界

Apollonian carpets and the boundary of the Kahler cone of a hyperkahler manifold

该研究探讨超Kähler流形丰沛锥的迷向边界与全纯自同构群动力学的关系，证明迷向边界上无理点均属极限集，实解析曲线含于测地球面，还研究了Apollonian地毯的分形几何及其与自同构群的关联。

Nef cones of projective bundles over surfaces and Seshadri constants

In this article, we give a description of the closed cone of curves of the projective bundle $\mathbb{P}(E)$ over a smooth projective variety $X$. Using duality, we then calculate the nef cone of divisors in $\mathbb{P}(E)$ over some special surfaces $X$ and for some special bundles on $X$. As an application, we also calculate the Seshadri constants of semistable ample vector bundles with vanishing discriminant on some special ruled surfaces at special points.

Extremal rays and automorphisms of holomorphic symplectic varieties

We survey recent results on ample cones and birational contractions of holomorphic symplectic varieties of K3 type, focusing on explicit constructions and concrete examples.

Nef cone and Seshadri constants on products of projective bundles over curves

Let $X = \mathbb{P}(E_1) \times_C \mathbb{P}(E_2)$ where $C$ is a smooth curve and let $E_1$, $E_2$ be vector bundles over $C$. In this paper, we extend the results in \cite{K-M-R} by computing the nef cone of $X$ without restriction on the rank or semistability of $E_1$ and $E_2$. We also study the Seshadri constants of ample line bundles on $X$. We calculate the Seshadri constants in some cases and give bounds in some of the remaining cases.

Nef cone volumes and discriminants of abelian surfaces

The nef cone volume appeared first in work of Peyre in a number-theoretic context on Del Pezzo surfaces, and it was studied by Derenthal and co-authors in a series of papers. The idea was subsequently extended to also measure the Zariski chambers of Del Pezzo surfaces. We start in this paper to explore the possibility to use this attractive concept to effectively measure the size of the nef cone on algebraic surfaces in general. This provides an interesting way of measuring in how big a space an ample line bundle can be moved without destroying its positivity. We give here complete results for…

Vanishing cohomology on a double cover

In this paper, we prove the irreducibility of the monodromy action on the anti-invariant part of the vanishing cohomology on a double cover of a very general element in an ample hypersurface of a complex smooth projective variety branched at an ample divisor. As an application, we study dominant rational maps from a double cover of a very general surface $S$ of degree$\geq 7$ in ${\mathbb P}^3$ branched at a very general quadric surface to smooth projective surfaces $Z$. Our method combines the classification theory of algebraic surfaces, deformation theory, and Hodge theory.

Effective cones of quotients of moduli spaces of stable n-pointed curves of genus zero

Let X_n := \bar M_{0,n}, the moduli space of n-pointed stable genus zero curves, and let X_{n,m} be the quotient of X_n by the action of the symmetric group S_{n-m} on the last n-m marked points. The cones of effective divisors of X_{n,m}, m = 0,1,2, are calculated. Using this, upper bounds for the cones Mov(X_{n,m}) generated by divisors with moving linear systems are calculated, m = 0,1, along with the induced bounds on the cones of ample divisors of \bar M_g and \bar M_{g,1}. As an application, the cone of effective divisors of \bar M_{2,1} is analyzed in detail.

Manifolds covered by lines and extremal rays

Let $X$ be a smooth complex projective variety and let $H \in \pic(X)$ be an ample line bundle. Assume that $X$ is covered by rational curves with degree one with respect to $H$ and with anticanonical degree greater than or equal to $(\dim X -1)/2$. We prove that there is a covering family of such curves whose numerical class spans an extremal ray in the cone of curves $\cone(X)$.

Unstable loci in flag varieties and variation of quotients

We consider the action of a semisimple subgroup $\hat G$ of a semisimple complex group $G$ on the flag variety $X=G/B$, and the linearizations of this action by line bundles $\mathcal L$ on $X$. The main result is an explicit description of the associated unstable locus in dependence of $\mathcal L$, as well as a combinatorial formula for its (co)dimension. We observe that the codimension is equal to 1 on the regular boundary of the $\hat G$-ample cone, and grows towards the interior in steps by 1, in a way that the line bundles with unstable locus of codimension $q$ form a convex polyhedral c…

Strong Stability of Cotangent Bundles of Cyclic Covers

Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$ of $\dim X\geq 4$ and Picard number $ρ(X)=1$. Suppose that $X$ satisfies $H^i(X,F^{m*}_X(\Omg^j_X)\otimes\Ls^{-1})=0$ for any ample line bundle $\Ls$ on $X$, and any nonnegative integers $m,i,j$ with $0\leq i+j<\dim X$, where $F_X:X\rightarrow X$ is the absolute Frobenius morphism. We prove that by procedures combining taking smooth hypersurfaces of dimension $\geq 3$ and cyclic covers along smooth divisors, if the resulting smooth projective variety $Y$ has ample (resp. nef) canonical bundle …

Rational curves on homogeneous cones

Let G/Q be an homogeneous variety embedded in a projective space P thanks to an ample line bundle L. Take a projective space containing P and form the cone X over G/Q, we call this a cone over an homogeneous variety. Let $α$ a class of 1-cycle on X. In this article we describe the irreducible components of the scheme of morphisms of class $α$ from a rational curve to X. The situation depends on the line bundle L : if the projectivised tangent space to the vertex contains lines (i.e. if G/Q contains lines in P) then the irreducible components are described as in our paper math.AG/0407123 by the…

On curves with high multiplicity on $\mathbb{P}(a,b,c)$ for $\min(a,b,c)\leq4$

On a weighted projective surface $\mathbb{P}(a,b,c)$ with $\min(a,b,c)\leq 4$, we compute lower bounds for the {\em effective threshold} of an ample divisor, in other words, the highest multiplicity a section of the divisor can have at a specified point. We expect that these bounds are close to being sharp. This translates into finding divisor classes on the blowup of $\mathbb{P}(a,b,c)$ that generate a cone contained in, and probably close to, the effective cone.

K3曲面的典范流与高度

Canonical currents and heights for K3 surfaces

本文在K3曲面丰沛锥的扩大边界上构造了自同构群等变的典范正流与高度连续族，并为椭圆纤维化曲面上的高度函数和流构造了优选代表元。

On the monodromy of irreducible symplectic manifolds

Exploiting recent results on the ample cone of irreducible symplectic manifolds, we provide a different point of view for the computation of their monodromy groups. In particular, we give the final step in the computation of the monodromy group for generalised Kummer manifolds and we prove that the monodromy of O'Grady's ten dimensional manifold is smaller than what was expected.

Involutions and linear systems on holomorphic symplectic manifolds

A $K3$ surface with an ample divisor of self-intersection 2 is a double cover of the plane branched over a sextic curve. We conjecture that a similar statement holds for the generic couple $(X,H)$ with $X$ a deformation of $(K3)^{[n]}$ and $H$ an ample divisor of square 2 for Beauville's quadratic form. If $n=2$ then according to the conjecture $X$ is a double cover of a (singular) sextic 4-fold in $\PP^5$. It follows from the conjecture that a deformation of $(K3)^{[n]}$ carrying a divisor (not necessarily ample) of degree 2 has an anti-symplectic birational involution. We test the conjecture…

Seshadri constants on the self-product of an elliptic curve

The purpose of this paper is to study Seshadri constants on the self-product $E\times E$ of an elliptic curve $E$. We provide explicit formulas for computing the Seshadri constants of all ample line bundles on the surfaces considered. As an application, we obtain a good picture of the behaviour of the Seshadri function on the nef cone.

射影空间乘积的 Calabi-Yau 完全交的双有理自同构群与可移动锥定理

Birational automorphism groups and the movable cone theorem for Calabi-Yau complete intersections of products of projective spaces

本文推广 Cantat–Oguiso 的 Coxeter 群方法，证明射影空间乘积中一般 Calabi-Yau 完全交的可移动锥猜想并刻画其双有理自同构群，还描述可移动锥边界及数值维数应用。

Nef divisors on $\bar{M}_{0,n}$ from GIT

We introduce and study the GIT CONE of $\bar{M}_{0,n}$, which is generated by the pullbacks of the natural ample line bundles on the GIT quotients $(\mathbb P^1)^n//SL(2)$. We give an explicit formula for these line bundles and prove a number of basic results about the GIT cone. As one application, we prove unconditionally that the log canonical models of $\bar{M}_{0,n}$ with a symmetric boundary divisor coincide with the moduli spaces of weighted curves or with the symmetric GIT quotient, extending the result of Matt Simpson arXiv:0709.4037. (Cf. also a different proof by Fedorchuk and Smyth …

The dual Kaehler cone of compact Kaehler threefolds

We consider a compact Kaehler manifold whose dual Kaehler cone contains a rational interior point. The general problem we have in mind is how far the manifold is from being projective; i.e. we ask for the algebraic dimension. We prove e.g. that if the interior point is represented by an effective curve in dimension 3, then the manifold has algebraic dimension at least 2 unless the variety is a so-called simple non-Kummer 3-fold. We also investigate 3-folds containing curves with ample normal bundle. It seems likely that examples with algebraic dimension 2 really exist.

连续爆发的相对曲线锥与极值收缩

Relative cone of curves and contractions of a successive blowup

本文证明沿具有简单正规交叉的子簇连续爆发的相对曲线锥由有限个初等曲线生成，且每个面可收缩，并描述极值收缩的例外轨迹及小收缩的D-翻转存在性。

Slope Semistability and Positive cones of Grassmann bundles

Let $E$ be a vector bundle of rank $r$ on a smooth complex projective variety $X$. In this article, we compute the nef and pseudoeffective cones of divisors in the Grassmann bundle $Gr_X(k,E)$ parametrizing $k$-dimensional subspaces of the fibers of $E$, where $1\leq k \leq rank(E)$, under assumptions on $X$ as well as on the vector bundle $E$. In particular, we show that nef cone and the pseudoeffective cone of $Gr_X(k,E)$ coincide if and only if $E$ is a slope semistable bundle on $X$ with $c_2(End(E))=0$. We also discuss about the nefness and ampleness of the universal quotient bundle $Q_k$…

关于奇变分的奇异品种III

On infinitesimal deformations of singular varieties III

本文研究了通过粘合三条射影线形成的奇异曲线的仿射锥，并证明其线丛非常量，从而嵌入到$\mathbb{P}^9$中。

Some geometric properties of Hilbert schemes of space curves

Let $H$ be the Hilbert scheme of curves in complex projective $3$-space, with $d\geq 3$ and genus $g \leq (d-2)^2/4$. A complete, explicit description of the cone of curves and the ample cone of $H$ is given. From this, partial results on the group $\mathop{Aut}(H)$ are deduced.

The cone conjecture for Calabi-Yau pairs in dimension two

We prove the Morrison-Kawamata cone conjecture for klt Calabi-Yau pairs in dimension 2. That is, for a large class of rational surfaces as well as K3 surfaces and abelian surfaces, the action of the automorphism group of the surface on the convex cone of ample divisors has a rational polyhedral fundamental domain. More concretely: there many be infinitely many curves with negative self-intersection on the surface, but all such curves fall into finitely many orbits under the automorphism group of the surface. The proof uses the geometry of groups acting on hyperbolic space. We deduce a characte…

单G-簇上的正性

Positivity on simple $G$-varieties

本文研究具有唯一闭轨道的单G-射影簇上B-等变向量丛的正性，证明丰富性与nef性可由有限条B-稳定曲线上的限制判定，并计算了汇点处爆破的nef锥及Seshadri常数。

Remarks on nef and movable cones of hypersurfaces in Mori dream spaces

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…

$g\leq 35$ 的对称 F-猜想

Symmetric F-conjecture for $g\leq 35$

本文证明了 $g\leq 35$ 时描述 $\overline{M}_{0,g}/S_g$ 和 $\overline{M}_g$ 丰富锥的对称 F-猜想。

Birational geometry of hypersurfaces in products of projective spaces

We study the birational properties of hypersurfaces in products of projective spaces. In the case of hypersurfaces in P^m x P^n, we describe their nef, movable and effective cones and determine when they are Mori dream spaces. Using these results, we give new simple examples of non-Mori dream spaces and analogues of Mumford's example of a strictly nef line bundle which is not ample.

Calabi-Yau threefolds of quotient type

First, we classify Calabi-Yau threefolds with infinite fundamental group by means of their minimal splitting coverings introduced by Beauville, and deduce that the nef cone is a rational simplicial cone and any rational nef divisor is semi-ample if the second Chern class is identically zero. We also derive a sufficient condition for the fundamental group to be finite in terms of the Picard number in an optimal form. Next, we give a concrete structure Theorem concerning $c_{2}$-contractions of Calabi-Yau threefolds as a generalisation and also a correction of our earlier works for simply connec…

射影空间乘积中 Calabi-Yau 完全交的体积

Volumes in Calabi-Yau Complete Intersection of Products of Projective Space

本文证明一般 Calabi-Yau 完全交的双有理自同构群具有 Lorentzian 性质，并利用 Kawamata-Morrison 锥定理计算小参数情形下的除子体积，同时给出涉及 digamma 函数的 Cartier 除子体积实例。

环面簇上Frobenius的几何

The geometry of Frobenius on toric varieties

本文通过定义Frobenius支撑及除子与1-闭链的F-有效性，揭示了F-有效曲线锥与Mori锥的相互作用如何刻画环面簇的极值Mori收缩，并证明Frobenius迹核丰沛当且仅当Picard秩为1。