Stefan Schroeer
Given an invertible sheaf, does it come from a Cartier divisor? This might fail in presence of embedded components. I give some examples and characterize those invertible sheaves that allow a Cartier divisor.
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In the previous paper [7], we introduced a notion of pairs of adelic R-Cartier divisors and R-base conditions. The purpose of this paper is to propose an extended notion of adelic R-Cartier divisors that we call an l1-adelic R-Cartier divisors, and to show that the arithmetic volume function defined on the space of pairs of l1-adelic R-Cartier divisors and R-base conditions is continuous along the directions of l1-adelic R-Cartier divisors.
Consider a one-parameter family of algebraic varieties degenerating to a reducible one. Our main result is a formula for the fundamental cycle of the limit subscheme of any family of effective Cartier divisors. The formula expresses this cycle as a sum of Cartier divisors, not necessarily effective, of the components of the limit variety.
The main goal of this paper is to show that the notions of Weil and Cartier $\mathbb{Q}$-divisors coincide for $V$-manifolds and give a procedure to express a rational Weil divisor as a rational Cartier divisor. The theory is illustrated on weighted projective spaces and weighted blow-ups.
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。
In this paper, we introduce positivity notions for pairs of adelic R-Cartier divisors and R-base conditions, and study fundamental properties of the arithmetic volumes defined for such pairs. We show that the Gateaux derivatives of the arithmetic volume function at big pairs along the directions of adelic R-Cartier divisors are given by suitable arithmetic positive intersection numbers. As a corollary, we obtain an Arakelov theoretic analogue of the Bonnesen-Diskant inequality in convex geometry.
Virtual Cartier divisors and blow-ups
本文基于“虚拟有效Cartier除子”概念证明正则浸入子概形胀开的泛性质，并在导出代数几何中构造拟光滑闭浸入的胀开，推进了胀开相关理论研究。
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).
Semigroups, Cartier divisors and convex bodies
本文通过几何和代数方法证明了Khovanskii-Teissier不等式，利用凸体和Cartier除子的交理论，概述了半群与凸体的相关理论。
A Note on the Strict Transformation of an Effective Cartier Divisor
本文研究有效 Cartier 除子的严格变换，给出沿覆盖其奇异轨迹的不交光滑子簇爆破后所得空间光滑的充分条件，并在维数条件下构造 Kuznetsov 意义下的弱范畴 crepant 解消。
We shall investigate flipping contractions from a semi-stable 4-fold $X$ whose degenerate fiber is a union of Cartier divisors which are terminal factorial 3-folds. Especially we shall prove that $X$ is smooth along the flipping locus, and that the flip exists for such contractions.
Effective calculation of local Weil functions via presentations of Cartier divisors
本文从 Cartier 除子表示出发，结合 Bombieri–Gubler 方法与既往工作，建立计算射影簇上局部 Weil 函数的理论有效算法。
In algebraic geometry, theorems of Küronya and Lozovanu characterize the ampleness and the nefness of a Cartier divisor on a projective variety in terms of the shapes of its associated Okounkov bodies. We prove the analogous result in the context of Arakelov geometry, showing that the arithmetic ampleness and nefness of an adelic $\mathbb{R}$-Cartier divisor $\overline{D}$ are determined by arithmetic Okounkov bodies in the sense of Boucksom and Chen. Our main results generalize to arbitrary projective varieties criteria for the positivity of toric metrized $\mathbb{R}$-divisors on toric varie…
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…
We generalize to all normal complex algebraic varieties the valuative characterization of multiplier ideals due to Boucksom-Favre-Jonsson in the smooth case. To that end, we extend the log discrepancy function to the space of all real valuations, and prove that it satisfies an adequate properness property, building upon previous work by Jonsson-Mustaţă. We next give an alternative definition of the concept of numerically Cartier divisors previously introduced by the first three authors, and prove that numerically Q-Cartier divisors coincide with Q-Cartier divisors for rational singularities. T…
Some divisor class relations are proved and used to compute the Cartier divisor class of the virtual canonical bundle for genus 0 maps to a smooth target.
We compute divisors class groups of singular surfaces. Most notably we produce an exact sequence that relates the Cartier divisors and almost Cartier divisors of a surface to the those of its normalization. This generalizes Hartshorne's theorem for the cubic ruled surface in P^3. We apply these results to limit the possible curves that can be set-theoretic complete intersection in P^3 in characteristic zero.
Generalizing cones over projective toric varieties, we present arbitrary toric varieties as quotients of quasiaffine toric varieties. Such quotient presentations correspond to groups of Weil divisors generating the topology. Groups comprising Cartier divisors define free quotients, whereas $\QQ$-Cartier divisors define geometric quotients. Each quotient presentation yields homogeneous coordinates. Using homogeneous coordinates, we express quasicoherent sheaves in terms of multigraded modules and describe the set of morphisms into a toric variety.
The birational classification of varieties inevitably leads to the study of singularities. The types of singularities that occur in this context have been studied by Mori, Kollár, Reid, and others, beginning in the 1980s with the introduction of the minimal model program. Normal singularities that are terminal, canonical, log terminal, and log canonical, and their non-normal counterparts, are typically studied using a resolution of singularities (or a semiresolution) and finding numerical conditions that relate the canonical class of the variety to that of its resolution. In order to do this, …
We consider the operation of intersecting with a locally principal Cartier divisor (i.e., a Cartier divisor which is principal on some neighborhood of its support). We describe this operation explicitly on the level of cycles and rational equivalences and as a corollary obtain a formula for rational equivalence between intersections of two locally principal Cartier divisors. Such canonical rational equivalence applies quite naturally to the setting of algebraic stacks. We present two applications: (i) a simplification of the development of Fulton-MacPherson-style intersection theory on Deligne…
The main object of the present paper is a numerical criterion determining when a Weil divisor of a $\Q$--factorial complete toric variety admits a positive multiple Cartier divisor which is either numerically effective (nef) or ample. It is a consequence of $\Z$--linear interpretation of Gale duality and se\-con\-dary fan as developed in several previous papers of us. As a byproduct we get a computation of the Cartier index of a Weil divisor and a numerical characterization of weak $\Q$--Fano, $\Q$--Fano, Gorenstein, weak Fano and Fano toric varieties. Several examples are then given and studi…
Let $(X, Δ)$ be a log pair in characteristic $p>0$ and $P$ be a (not necessarily closed) point of $X$. We show that there exists a constant $δ>0$ such that $τ(X, Δ)_P= τ(X, Δ+ D)_P$ for each effective $\mathbb{Q}$-Cartier divisor $D$ with $\mathrm{mult}_P(D) <δ$. As its application, we show that if $D$ is an $\mathbb{R}$-Cartier divisor on a strongly $F$-regular projective variety, then the non-nef locus of $D$ coincides with the restricted base locus of $D$. This is a generalization of a result of Mustaţǎ to the singular case and can be viewed as a characteristic $p$ analogue of a result of C…
This note describes a Macaulay2 package for handling divisors. Group operations for divisors are included. There are methods for converting divisors to reflexive or invertible sheaves. Additionally, there are methods for checking whether divisors are Cartier, $Q$-Cartier, simple normal crossings, generate base point free linear systems, or satisfy numerous other conditions.
For a smooth projective variety $P$, we construct a Cartier divisor supported on the incidence locus in $\mathscr{C}_a (P) \times \mathscr{C}_{\dim(P)-a-1}(P)$. There is a natural definition of the corresponding line bundle on a product of Hilbert schemes, and we show this bundle descends to the Chow varieties. This answers a question posed by Mazur.
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 normal projective variety X is called Fano if a multiple of the anticanonical Weil divisor, -K_X, is an ample Cartier divisor, the index of a Fano variety is the number i(X):=sup{t: -K_X= tH, for some ample Cartier divisor H}. Mukai announced, the classification of smooth Fano manifolds X of index i(X)=n-2, under the assumption that the linear system |H| contains a smooth divisor. In this paper we prove that this assumption is always satisfied. Therefore the result of Mukai provide a complete classification of smooth Fano n-folds of index $i(X)=n-2$, Mukai manifolds.
The volume of a Cartier divisor on a projective variety is a nonnegative real number that measures the asymptotic growth of sections of multiples of the divisor. It is known that the set of these numbers is countable and has the structure of a multiplicative semigroup. At the same time it still remains unknown which nonnegative real algebraic numbers arise as volumes of Cartier divisors on some variety. Here we extend a construction first used by Cutkosky, and use the theory of real multiplication on abelian varieties to obtain a large class of examples of algebraic volumes. We also show that …
Parametrization of positroid cells by Dubrovin-Natanzon Cartier divisors on maximal Mumford curves
该研究证明全正Schubert胞是对应Le图的极大Mumford曲线雅可比簇的实分支，并构造了正合胞点与该曲线上Dubrovin-Natanzon除子间的双射。
We first introduce a weak type of Zariski decomposition in higher dimensions: an $\R$-Cartier divisor has a weak Zariski decomposition if birationally and in a numerical sense it can be written as the sum of a nef and an effective $\R$-Cartier divisor. We then prove that there is a very basic relation between Zariski decompositions and log minimal models which has long been expected: we prove that assuming the log minimal model program in dimension $d-1$, a lc pair $(X/Z,B)$ of dimension $d$ has a log minimal model if and only if $K_X+B$ has a weak Zariski decomposition$/Z$.
In an earlier paper we constructed a Cartier divisor on the theta divisor of a principally polarised abelian variety whose support is precisely the ramification locus of the Gauss map. In this note we discuss a Green's function associated to this locus. For jacobians we relate this Green's function to the canonical Green's function of the corresponding Riemann surface.
The comparison of the 0-th Suslin homology and Chow groups of 0-cycles with modulus
本文证明了在正特征域上，射影正规曲面及任意维数射影光滑概形中，带模零循环周群与0次Suslin同调之间的同构关系。
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.
The effectivity up to R-linear equivalence (Dirichlet property) of pseudoeffective adelic R-Cartier divisors is a subtle problem in arithmetic geometry. In this article, we propose sufficient conditions for the Dirichlet property by using the dynamic system in the classic Arakelov geometry setting. We also give a numerical criterion of the Dirichlet property for adelic R-Cartier divisors on curves over a trivially valued field.
Let $X$ be a normal projective variety. A surjective endomorphism $f:X\to X$ is int-amplified if $f^\ast L - L =H$ for some ample Cartier divisors $L$ and $H$. This is a generalization of the so-called polarized endomorphism which requires that $f^*H\sim qH$ for some ample Cartier divisor $H$ and $q>1$. We show that this generalization keeps all nice properties of the polarized case in terms of the singularity, canonical divisor, and equivariant minimal model program.
Let $X$ be a complete normal variety, $B$ an effective $\mathbb{R}$-divisor on $X$, and $D$ a Cartier divisor on $X$. Assume that the pair $(X, B)$ is log terminal. We consider the problem whether $H^0(X, D) \ne 0$ and obtain some results in lower dimensions.
Generalized Minkowski weights and Chow rings of T-varieties
本文用广义 Minkowski 权重组合刻画了复杂度为一的有理 T-簇的 Fulton 操作 Chow 上同调群，并用组合数据描述与不变 Cartier 除子的交积，给出计算最高交截数的新方法。
We investigate properties of potentially Du Bois singularities, that is, those that occur on the underlying space of a Du Bois pair. We show that a normal variety $X$ with potentially Du Bois singularities and Cartier canonical divisor $K_X$ is necessarily log canonical, and hence Du Bois. As an immediate corollary, we obtain the Lipman-Zariski conjecture for varieties with potentially Du Bois singularities. We also show that for a normal surface singularity, the notions of Du Bois and potentially Du Bois singularities coincide. In contrast, we give an example showing that in dimension at leas…
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 prove an analogue of the Lefschetz (1,1) Theorem characterizing cohomology classes of Cartier divisors (or equivalently first Chern classes of line bundles) in the second integral cohomology. Let $X$ be a normal complex projective variety. We show that the classes of Cartier divisors in $H^2(X,Z)$ are precisely the classes $x$ such that (i) the image of $x$ in $H^2(X,C)$ (cohomology with complex coefficients) lies in $F^1 H^2(X,C)$ (first level of the Hodge filtration for Deligne's mixed Hodge structure), and (ii) $x$ is Zariski-locally trivial, i.e., there is a covering of $X$ by Zariski o…
In this study, we give an alternative and elementary proof to Tsuji's criterion for a Cartier divisor to be numerically trivial.
We introduce an adelic Cartier divisor over a trivially valued field and discuss the bigness of it. For bigness, we give the integral representation of the arithmetic volume and prove the existence of limit of it. Moreover, we show that the arithmetic volume is continuous and log concave.
We investigate the vanishing of $H^1(X,\mathcal{O}_X(-D))$ for a big and nef $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor $D$ on a log del Pezzo surface $(X,Δ)$ over an algebraically closed field of positive characteristic $p$.
Let $(X,Δ)$ be a proper dlt pair and $L$ a nef Cartier divisor such that $aL-(K_X+Δ)$ is nef and log big on $(X,Δ)$ for some $a\in {\mathbb Z}_{>0}$. Then $|mL|$ is base point free for every $m\gg 0$.
A contraction theorem for divisors fibering over a curve
本文针对纤维化到曲线上的 Q-Cartier 除子，给出了双亚缅收缩存在的充分条件，并作为推论恢复了紧 Kähler 三维簇的收缩结果。
Divisors on coherent schemes and homogeneous spaces
本文研究相干基上齐性空间上除子的正性与可延展性，发展了局部相干概形上的估值除子理论，建立了有效估值除子与秩为一的反射sheaf的精确对应关系，推导出非Noetherian的Ramanujam-Samuel定理。
Let f : X --> Y be a projective morphism of smooth algebraic varieties over an algebraically closed field of characteristic zero with dim f = 1. Let E be a vector bundle of rank r on X. In this paper, we would like to show that if X_y is smooth and E_y is semistable for some point y of Y, then f_* (2r c_2(E) - (r-1) c_1(E)^2) is weakly positive at y. We apply this result to obtain the following description of the cone of weakly positive $\QQ$-Cartier divisors on the moduli space of stable curves. Let M_g (resp. M_g^0) be the moduli space of stable (resp. smooth) curves of genus g >= 2. Let h b…
Let $X$ be a projective variety over an algebraically closed field $k$ of arbitrary characteristic $p \ge 0$. A surjective endomorphism $f$ of $X$ is $q$-polarized if $f^\ast H \sim qH$ for some ample Cartier divisor $H$ and integer $q > 1$. Suppose $f$ is separable and $X$ is $\mathbb{Q}$-Gorenstein and normal. We show that the anti-canonical divisor $-K_X$ is numerically equivalent to an effective $\mathbb{Q}$-Cartier divisor, strengthening slightly the conclusion of Boucksom, de Fernex and Favre (Theorem C) and also covering singular varieties over an algebraically closed field of arbitrary…
Using the theory of Frobenius singularities, we show that 13mK_X + 45mA is very ample for an ample Cartier divisor A on a Kawamata log terminal surface X with Gorenstein index m, defined over an algebraically closed field of characteristic p>5.
On boundedness of indices of minimal pairs -- surfaces
本文研究射影 klt 对的指标有界性，证明在给定维数和 Cartier 指标时，无法仅由这些参数确定一个整数使得线性等价成立。
Counterexamples to the Ambro--Kawamata effective non-vanishing conjecture
本文在特征0或p≠3的代数闭域上构造反例，否证Ambro--Kawamata有效非消失猜想，并给出相关几何变体与阶5 Jacobian商。
In this paper we study a notion of volume for Cartier divisors on arbitrary blow-ups of normal complex algebraic varieties of dimension greater than one, with a distinguished point. We apply this to study a volume for normal isolated singularities, generalizing work on surfaces done by Wahl. We also compare this volume of isolated singularities to a different generalization due to Boucksom, de Fernex and Favre.
We prove two theorems on the locally finite decompositions of the cones of divisors by the cones which correspond to canonical and minimal models. We introduce the concept of the numerical linear systems in order to simplify the argument on the Zariski decompositions.
Cyclic covers via mixed Hodge modules
本文针对复代数簇上由线丛整体截面定义的有效卡蒂埃除子，用混合霍奇模理论描述其N重循环覆盖的构造方法。
We study the cones of q-ample divisors on smooth complex varieties. In favourable cases, we identify a part where the closure of this cone and the nef cone have the same boundary. This is especially interesting for Fano (or almost Fano) varieties.
We show that for a smooth projective variety $X$ over a field $k$ and a reduced effective Cartier divisor $D \subset X$, the Chow group of 0-cycles with modulus $\mathrm{CH}_0(X|D)$ coincides with the Suslin homology $H^S_0(X \setminus D)$ under some necessary conditions on $k$ and $D$. We derive several consequences, and we answer to a question of Barbieri-Viale and Kahn.
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.
The number of smooth varieties in an MMP on a 3-fold of Fano type
本文证明对于 Fano 型三维簇 X 及其上的可动 Q-Cartier Weil 除子 D，执行 D-MMP 过程中出现的光滑簇的数量不超过 1 + h^1(X, 2D)，并证明了 Kodaira 消失定理对于 Fano 型三维簇上可动除子的部分逆命题。
A generalized Euler sequence over a complete normal variety X is the unique extension of the trivial bundle V \otimes O_X by the sheaf of differentials Ω_X, given by the inclusion of a linear space V in Ext^1(O_X,Ω_X). For Λ, a lattice of Cartier divisors, let R_Λdenote the corresponding sheaf associated to V spanned by the first Chern classes of divisors in Λ. We prove that any projective, smooth variety on which the bundle R_Λsplits into a direct sum of line bundles is toric. We describe the bundle R_Λin terms of the sheaf of differentials on the characteristic space of the Cox ring, provide…
We show that conformal blocks divisors of type B_r and D_r at level one are effective sums of boundary divisors of $\bar{M}_{0,n}$. We also prove that the conformal blocks divisor of type $B_r$ at level 1 with weights (ω_1,\dots,ω_1) scales linearly with the level.
Remarks on Milnor K-theory and Tate's conjecture for divisors
本文建立了有限域上除子 Tate 猜想与代数闭包上三变量函数域第三 Milnor K-群的一个明确代数问题之间的等价关系。
The Tate conjecture for divisors on varieties over number fields is equivalent to finiteness of $\ell$-primary torsion in the Brauer group. We show that this finiteness is actually uniform in one-dimensional families for varieties that satisfy the Tate conjecture for divisors -- e.g. abelian varieties and $K3$ surfaces.
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