Frédéric Campana, Jungkai A. Chen, Thomas Peternell
Let $X$ be a projective manifold of dimension $n$ and $L$ a strictly nef line bundle on $X$. Then $K_X+tL$ is ample if $t > n+1$ in the following cases. 1.) $\text{dim} X = 3$ unless (possibly) $X$ is a Calabi-Yau with $c_2 \cdot L=0$; 2.) $κ(X) \ge n-2$; 3.) $\text{dim} α(X) \ge n-2$, with $α: X \to A$ the Albanese map.
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We give a direct proof, valid in arbitrary characteristic, of nefness for two families of F-nef divisors on $\bar{M}_{0,n}$. The divisors we consider include all type A level one conformal block divisors as well as divisors previously not known to be nef.
We prove a structure theorem for projective varieties with nef anticanonical divisors.
In this paper we study normal surfaces whose anticanonical divisors are strictly nef, i.e. (-K)C>0 for every curve C.
We describe a nef divisor $D_P$ on $\bar{M}_{0,12}$ that is not numerically equivalent to an effective sum of boundary divisors.
Let $X$ be a smooth projective rationally connected threefold with nef anticanonical divisor. We give a classification for the case when $-K_X$ is not semi-ample.
We study lc pairs polarized by a nef and log big divisor. After proving the minimal model theory for projective lc pairs polarized by a nef and log big divisor, we prove the effectivity of the Iitaka fibrations and some boundedness results for dlt pairs polarized by a nef and log big divisor.
In this paper, we give a numerical characterization of nef arithmetic R-Cartier divisors of C^0-type on an arithmetic surface. Namely an arithmetic R-Cartier divisor D of C^0-type is nef if and only if D is pseudo-effective and deg(D^2) = vol(D).
Strictly nef divisors on K-trivial fourfolds
本文证明了K-平凡四维流形上严格nef除子的丰度猜想和Serrano猜想，指出具有平凡典范丛和消失非正则性的射影四维流形上的严格nef除子都是丰沛的。
We study the Iitaka-Kodaira dimension of nef relative anti-canonical divisors. As a consequence, we prove that given a complex projective variety with klt singularities, if the anti-canonical divisor is nef, then the dimension of a general fibre of the maximal rationally connected fibration is at least the Iitaka-Kodaira dimension of the anti-canonical divisor.
Let $X$ be a smooth complex projective rationally connected threefold with $-K_X$ nef and not semi-ample. In our previous work, we classified all such threefolds when $|{-}K_X|$ has no fixed divisor. In this paper, we continue our classification when $|{-}K_X|$ has a non-zero fixed divisor.
In this paper, we classify projective toric birational morphisms from Gorenstein toric 3-folds onto the 3-dimensional affine space with relatively ample anti-canonical divisors.
Under some conditions on the deformation type, which we expect to be satisfied for arbitrary irreducible symplectic varieties, we describe which big and nef line bundles on irreducible symplectic varieties have base divisors. In particular, we show that such base divisors are always irreducible and reduced. This is applied to understand the behaviour of divisorial base components of big and nef line bundles under deformations and for K3$^{[n]}$-type and Kum$^n$-type.
We show that $sl_2$ conformal block divisors do not cover the nef cone of $\bar{M}_{0,6}$, or the $S_9$-invariant nef cone of $\bar{M}_{0,9}$. A key point is to relate the nonvanishing of intersection numbers between these divisors and F-curves to the nonemptiness of some explicitly defined polytopes. Several experimental results and some open problems are also included.
In this paper, we study the structure of projective space bundles whose relative anti-canonical line bundle is nef. As an application, we get a characterization of abelian varieties up to finite etale covering.
In this paper, we use canonical bundle formulas to prove some generalizations of an old theorem of Kawamata on the semiampleness of nef and abundant log canonical divisors. In particular, we show that for klt pairs $(X,B)$ with $K_X+B$ effective, $L \in Pic (X)$ nef, nefness and abundance of $K_X+B+L$ implies semiampleness. This essentially generalizes Kawamata's theorem to the setting of generalized abundance.
In this paper we study smooth projective rational surfaces, defined over an algebraically closed field of any characteristic, with pseudo-effective anticanonical divisor. We provide a necessary and sufficient condition in order for any nef divisor to be semiample. We adopt our criterion to investigate Mori dream surfaces in the complex case.
We give a criterion for a nef divisor $D$ to be semiample on a Calabi--Yau threefold $X$ when $D^3=0=c_2(X)\cdot D$ and $c_3(X)\neq 0$. As a direct consequence, we show that on such a variety $X$, if $D$ is strictly nef and $ν(D)\neq 1$, then $D$ is ample; we also show that if there exists a nef non-ample divisor $D$ with $D\not\equiv 0$, then $X$ contains a rational curve when its topological Euler characteristic is not $0$.
A question on effective strictly nef divisors (with an appendix by Andreas Höring)
该研究提出射影簇上有效严格nef Cartier除子是否均为大除子的问题，Andreas Höring在附录中构造反例给出了否定答案。
A characterization of nef and good divisors is given: a divisor D on a smooth complex projective variety is nef and good if and only if the asymptotic multiplier ideals of sufficiently high multiples of e(D) D$ are trivial, where e(D) denotes the exponent of the divisor D. Some results of the same kind are proved in the analytic setting.
We prove that any nef b-divisor class on a projective variety defined over an algebraically closed field of characteristic 0 is a decreasing limit of nef Cartier classes. Building on this technical result, we construct an intersection theory of nef b-divisors, and prove several variants of the Hodge index theorem inspired by the work of Dinh and Sibony. We show that any big and basepoint free curve class is a power of a nef b-divisor, and relate this statement to Zariski decompositions of curves classes introduced by Lehmann and Xiao. Our construction allows us to relate various Banach spaces …
Let $X$ be a normal projective threefold with mild singularities, and $L_X$ a strictly nef $\mathbb{Q}$-divisor on $X$. First, we show the ampleness of $K_X+tL_X$ with sufficiently large $t$ if either the Kodaira dimension $κ(X)\neq 0$ or the augmented irregularity $q^{\circ}(X)\neq 0$. Second, we show that, if $(X,Δ)$ is a projective klt threefold pair with the anti-log canonical divisor $-(K_X+Δ)$ being strictly nef, then $X$ is rationally connected.
Any ample Cartier divisor D on a projective variety X is strictly nef (i.e. D.C>0 for any effective curve C on X). In general, the converse statement does not hold. But this is conjectured to be true for anticanonical divisors. The present paper establishes this fact for normal complex projective threefolds with canonical singularities. This result extends several previously known special cases. The proof rests mainly on sophisticated techniques of three dimensional birational geometry developed in the last two decades.
Serrrano's Conjecture says that if $L$ is a strictly nef line bundle on a smooth projective variety $X$, then $K_X+tL$ is ample for $ t > dim X+1$. In this paper I will prove a few cases of this conjecture. I will also prove a generalized version of this conjecture (due to Campana, Chen and Peternell) for surfaces. In the last section, assuming the SHGH conjecture, I will give a series of examples of strictly nef non ample divisors on surfaces of arbitary Kodaira dimension.
To a generically big adelic divisor, we can associate an arithmetic Okounkov body, which is a pair of the geometric Okounkov body and the concave transform of the Green functions. In this paper, we show that the infimum of the concave transform is given by the absolute minimum provided that the divisor is vertically nef. This is a partial generalization of results of Moriwaki (in the curve case) and of Burgos Gil-Moriwaki-Philippon-Sombra (in the toric case).
On a projective surface it is well-known that the set of curves orthogonal to a nef line bundle is either finite or uncountable. We show that this dichotomy fails in higher dimension by constructing a nef line bundle on a threefold which is trivial on countably infinitely many curves. This answers a question of Totaro. As a pleasant corollary, we exhibit a quasi-projective variety with only a countably infinite set of complete, positive-dimensional subvarieties.
Varieties with nef anticanonical divisors and Albanese morphisms of relative dimension one in positive characteristic
本文研究正特征代数闭域上具有 nef 反典范除数的光滑射影簇，并在其到 Albanese 簇的态射相对维数为一时给出精确结构。
We reduce the Abundance Conjecture in dimension 4 to the following numerical statement: if the canonical divisor K is nef and has maximal nef dimension, then K is big. From this point of view, we ``classify'' in dimension 2 nef divisors which have maximal nef dimension, but which are not big.
We show that a nef and big line bundle whose adjoint bundle has non-zero global sections on a nonsingular toric weak Fano 3-fold is normally generated. As a consequence, we see that all ample line bundles on a nonsingular toric weak Fano 3-fold are normally generated. We also show that an ample line bundle whose adjoint bundle has non-zero global sections on a Gorensein toric Fano 3-fold is normally generated. In our proof we do not use full classifications of Fano polytopes.
Minimal compactifications of the affine plane with nef canonical divisors
本文研究仿射平面的极小解析紧化，证明具有 nef 典范除子的此类紧化必含无理奇点，与先前对数典范情形形成对比。
Let M_g be the moduli space of smooth curves of genus g >= 3, and \bar{M}_g the Deligne-Mumford compactification in terms of stable curves. Let \bar{M}_g^{[1]} be an open set of \bar{M}_g consisting of stable curves of genus g with one node at most. In this paper, we determine the necessary and sufficient condition to guarantee that a Q-divisor D on \bar{M}_g is nef over \bar{M}_g^{[1]}, that is, (D . C) >= 0 for all irreducible curves C on \bar{M}_g with C \cap \bar{M}_g^{[1]} \not= \emptyset.
Geometric singularities of regular surfaces with nef anti-canonical divisors over imperfect fields
研究正则投影曲面在特征p≥7的域上，当反canonical除子非负且局部环为域时，曲面在域上几何积分的性质。
We show that the tangent bundle of a projective manifold with nef anticanonical class is generically nef. That is, its restriction to a curve cut out by general sufficiently ample divisors is a nef vector bundle. This confirms a conjecture of Peternell. As a consequence, the second Chern class of such a manifold has non-negative intersections with ample divisors. We also investigate under which conditions these positivities are strict, and answer a question of Yau.
In this short note we will show that every homogeneous strictly nef vector bundle on a complex flag variety is ample. Following this, we consider whether ampleness of a bundle on an abelian variety can be tested on curves.
Kawamata--Miyaoka type inequality for varieties with nef anticanonical divisor
针对n≥2的Q-因子ε-lc弱Fano簇，证明了满足特定系数约束的川又-宫冈型不等式，丰富了代数几何中Fano簇的相关理论。
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 …
We generalize the intersection theory of nef toric (Weil) b-divisors on smooth and complete toric varieties to the case of smooth and complete toroidal embeddings. As a key ingredient we show the existence of a limit measure, supported on the weakly embedded rational conical polyhedral space attached to the toroidal embedding, which arises as a limit of discrete measures defined via tropical intersection theory on the polyhedral space. We prove that the intersection theory of nef toroidal Cartier b-divisors can be extended continuously to nef toroidal Weil b-divisors and that their degree can …
We give another alternative proof to the Kawamata semiampleness theorem for the log canonical divisors on klt varieties which are nef and abundant. After the first version of this article was posted to the e-print Arxiv, Prof. Fujino notified the author that the quick and essential proof ([Fujino. On Kawamata's theorem.(EMS 2011), Rem 2.7]) is already known. The author would like to thank him. More precisely, Prof. Fujino already gave the quick and essential proof ([Fujino. On Kawamata's thm.(EMS 2011), Rem 2.7], [Fujino. Finite generation of the lc ring in dim 4. (Kyoto J. Math. 50 (2010)), R…
Threefolds of globally $F$-regular type with nef anti-canonical divisor
本研究针对Schwede和Smith的猜想特例，证明具nef反典范除子的光滑复射影三维簇若为整体$F$-正则型则是弱Fano簇，推进了相关代数几何猜想的验证。
Automorphism groups and Homogeneous fibrations on projective varieties with nef anticanonical divisors
本文研究具有 nef 反典范除子的代数簇的 Albanese 态射与极大有理链连通纤维化的齐次结构，构造自然群同态诱导 Aut^0 的 Chevalley 分解（相差等源），并类比 Nishi-Matsumura 定理，用现代语言重证之。
In [BM14b], the first author and Macrì constructed a family of nef divisors on any moduli space of Bridgeland-stable objects on a smooth projective variety X. In this article, we extend this construction to the setting of any separated scheme Y of finite type over a field, where we consider moduli spaces of Bridgeland-stable objects on Y with compact support. We also show that the nef divisor is compatible with the polarising ample line bundle coming from the GIT construction of the moduli space in the special case when Y admits a tilting bundle and the stability condition arises from a θ-stab…
Boucksom, Favre and Jonsson establish in [4] an analog of Diskant's inequality in convex geometry for nef and big line bundles on a complete algebraic variety over an algebraically closed field of characteristic zero (Theorem F [4]), from which they deduce a solution (Theorem D [4]) to a problem of Teissier (Problem B [29]) on proportionality of nef divisors. In this paper we extend these results to a complete variety over an arbitrary field k.
Let $(X, ω)$ be an n-dimensional compact Kähler manifold. Let $D=\sum (1-β_j) Y_j=\sum (1-β_j) [s_j=0]$ a divisor with simple normal crossings with $β_j \in ]0,1[$ such that $-(K_X+D)$ is nef. We show that its Albanese map is submersion outside an analytic set of codimension larger than two with connected fibres.
In these notes we investigate the cone of nef curves of projective varieties, which is the dual cone to the cone of pseudo-effective divisors. We prove a structure theorem for the cone of nef curves of projective $\mathbb Q$-factorial klt pairs of arbitrary dimension from the point of view of the Minimal Model Program. This is a generalization of Batyrev's structure theorem for the cone of nef curves of projective terminal threefolds.
We show that the augmented base locus coincides with the exceptional locus (i.e. null locus) for any nef $\mathbb{R}$-Cartier divisor on any scheme projective over a field (of any characteristic). Next we prove a semi-ampleness criterion in terms of the augmented base locus generalizing a result of Keel. We also study nef divisors with positive top intersection number, and discuss some problems related to augmented base loci of log divisors.
Splitting of algebraic fiber spaces with nef relative anti-canonical divisor and decomposition of $F$-split varieties
本文证明在完美域上具有nef相对反典范除子的代数纤维空间经有限覆盖基变换后分裂为乘积，并推广了正特征Beauville-Bogomolov分解，应用于丰沛猜想与有理点问题。
Let X be a projective irreducible symplectic manifold and L a non trivial nef divisor on X. Assume that the nef dimension of L is strictly less than the dimension of X. We prove that L is semiample
Nef divisors of surfaces given by pencils at infinity
本文研究了通过在复射影平面的特定点集上吹胀得到的曲面的 nef 结构和 Cox 环生成性，提出了关于纤维基点和自由点的生成元理论，并证明了 Cox 环的有限生成性条件。
Miyaoka type inequality for terminal threefolds with nef anti-canonical divisors
本文研究带nef反典范除子的终端射影3-簇的Miyaoka型不等式，给出陈类乘积的下界及紧性结果，通过部分分类与例子完成证明，还拓展了终端弱Fano簇的相关非零性与Miyaoka–Kawamata型不等式研究。
In this paper, we prove that given a flat generically smooth morphism between smooth projective varieties with $F$-pure closed fibers, if the source space is Fano, weak Fano or a variety with the nef anti-canonical divisor, then so is the target space. We also show that relative anti-canonical divisors of generically smooth surjective nonconstant morphisms are not nef and big in arbitrary characteristic.
Let $X$ be a complete algebraic variety over {\bf C}. We consider a log variety $(X,Δ)$ that is weakly Kawamata log terminal. We assume that $K_X+Δ$ is a {\bf Q}-Cartier {\bf Q}-divisor and that every irreducible component of $\lfloor Δ\rfloor$ is {\bf Q}-Cartier. A nef and big Cartier divisor $H$ on $X$ is called {\it nef and log big} on $(X,Δ)$ if $H |_B$ is nef and big for every center $B$ of non-"Kawamata log terminal" singularities for $(X,Δ)$. We prove that, if $L$ is a nef Cartier divisor such that $aL-(K_X+Δ)$ is nef and log big on $(X,Δ)$ for some $a \in$ {\bf N}, then the complete li…
Strictly nef divisors on singular threefolds
本文研究奇异三重簇上 Serrano 猜想，证明在 $\mathbb{Q}$-factorial Gorenstein terminal 条件下，除弱 Calabi–Yau 且第二陈类相交为零的情形外，$K_X+tL_X$ 对充分大的 $t$ 丰饶。
Positivity of the tangent bundle of rational surfaces with nef anticanonical divisor
本文研究了具有nef反典范除子的光滑射影有理曲面切丛的bigness性质，证明了有理椭圆曲面的切丛不是大的，并利用$(-2)$-曲线配置完全或部分确定了弱del Pezzo曲面切丛的bigness。
In this paper, we show that if the tangent bundle of a smooth projective variety is strictly nef, then it is isomorphic to a projective space; if a projective variety $X^n$ $(n>4)$ has strictly nef $Λ^2 TX$, then it is isomorphic to $\mathbb{P}^n$ or quadric $\mathbb{Q}^n$. We also prove that on elliptic curves, strictly nef vector bundles are ample, whereas there exist Hermitian flat and strictly nef vector bundles on any smooth curve with genus $g\geq 2$.
We prove that the non-nef locus and the restricted base locus of a pseudoeffective divisor coincide on KLT pairs. We also extend to KLT pairs F. Russo's characterization of nef and abundant divisors by means of asymptotic multiplier ideals.
Using currents with minimal singularities, we construct pointwise minimal multiplicities for a real pseudo-effective $(1,1)$-class $α$ on a compact complex $n$-fold $X$, which are the local obstructions to the numerical effectivity of $α$. The negative part of $α$ is then defined as the real effective divisor $N(α)$ whose multiplicity along a prime divisor $D$ is just the generic multiplicity of $α$ along $D$, and we get in that way a divisorial Zariski decomposition of $α$ into the sum of a class $Z(α)$ which is nef in codimension 1 and the class of its negative part $N(α)$, which is exceptio…
In this paper, we study smooth complex projective varieties $X$ such that some exterior power $\bigwedge^r T_X$ of the tangent bundle is strictly nef. We prove that such varieties are rationally connected. We also classify the following two cases. If $T_X$ is strictly nef, then $X$ isomorphic to the projective space $\mathrm{P}^n$. If $\bigwedge^2 T_X$ is strictly nef and if $X$ has dimension at least $3$, then $X$ is either isomorphic to $\mathrm{P}^n$ or a quadric $\mathrm{Q}^n$.
As a consequence of our recently established generalized Schmidt's subspace theorem for closed subschemes in general position, we prove a degeneracy theorem for integral points on the complement of a union of nef effective divisors. A novel aspect of our result is the attainment of a strong degeneracy conclusion (arithmetic quasi-hyperbolicity) under weak positivity assumptions on the divisors. The proof hinges on applying our recent theorem with a well-situated ample divisor realizing a certain lexicographical minimax. We also explore the connections with earlier work by other authors and mak…
In this note, we give a brief exposition on the differences and similarities between strictly nef and ample vector bundles, with particular focus on the circle of problems surrounding the geometry of projective manifolds with strictly nef bundles.
We determine the cone of nef divisors on the Voronoi compactification A_g^* of the moduli space A_g of principally polarized abelian varieties of dimension g for genus g=2,3. As a corollary we obtain that the spaces A_g^*(n) with level-n structure are a minimal, resp. canonical, model for g=2, n>=4, resp. n>=5 and g=3, n>=3, resp. n>=4. We give two proofs: The easy and quick one reduces the problem to \bar M_g where we can use a result of Faber. This approach cannot be generalized to higher genus g. The main point of the paper is, therefore, to give a second proof using theta functions and a r…
Let (X,H) be a polarized, smooth, complex projective surface, and let v be a Chern character on X with positive rank and sufficiently large discriminant. In this paper, we compute the Gieseker wall for v in a slice of the stability manifold of X. We construct explicit curves parameterizing non-isomorphic Gieseker stable sheaves that become S-equivalent along the wall. As a corollary, we conclude that if there are no strictly semistable sheaves of character v, the Bayer-Macri divisor associated to the wall is a boundary nef divisor on the moduli space of sheaves M_H(v). We recover previous resu…
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