Let X\subset PP^n be a projective scheme over a field, and let phi:X --> Y be a finite morphism. Our main result is a formula in terms of global data for the maximum of the Castelnuovo-Mumford regularity of the fibers of ϕ, considered as subschemes of \PP^n.
From an algebraic point of view, our formula is related to the theorem of Cutkosky-Herzog-Trung and Kodiyalam showing that for any homogeneous ideal I in a standard graded algebra S, the regularity of I^t can be written as dt+εfor some non-negative integers d, ε, and all large t. In the special case where I contains a power of S_+ and is generated by forms of a single degree, our formula gives an interpretation of ε: it is one less than the maximum regularity of a fiber of the morphism associated to I.
These formulas have strong consequences for ideals generated by generic forms.
We study the fibers of a projective morphism and some related algebraic problems. We characterize the analytic spread of a homogeneous ideal through properties of its syzygy matrix. Powers of linearly presented ideals need not be linearly presented, but we identify a weaker linearity property that is preserved under taking powers.
We first provide here a very short proof of a refinement of a theorem of Kodiyalam and Cutkosky, Herzog and Trung on the regularity of powers of ideals. This result implies a conjecture of Hà and generalizes a result of Eisenbud and Harris concerning the case of ideals primary for the graded maximal ideal in a standard graded algebra over a field. It also implies a new result on the regularities of powers of ideal sheaves. We then compare the cohomology of the stalks and the cohomology of the fibers of a projective morphism to the effect of comparing the maximum over fibers and over stalks of …
In this paper, we study the Albanese morphisms in positive characteristic. We prove that the Albanese morphism of a variety with nef anti-canonical divisor is an algebraic fiber space, under the assumption that the general fiber is $F$-pure. Furthermore, we consider a notion of $F$-splitting for morphisms, and investigate it of the Albanese morphisms. We show that an $F$-split variety has $F$-split Albanese morphism, and that the $F$-split Albanese morphism is an algebraic fiber space. As an application, we provide a new characterization of abelian varieties.
If a K3 surface is a fiber of a flat projective morphisms over a connected noetherian scheme over the complex number field, then any smooth connected fiber is also a K3 surface. Observing this, Professor Nam-Hoon Lee asked if the same is true for higher dimensional Calabi-Yau fibers. We shall give an explicit negative answer to his question as well as a proof of his initial observation.
Let k be an algebraically closed field and let X be a separated scheme of finite type over k of pure dimension d. We study the structure of the fibres of the truncation morphisms from the arc space of X to jet spaces of X and also between jet spaces. Our results are generalizations of results of Denef, Loeser, Ein and Mustata. We will use them to find the optimal lower bound for the poles of the motivic zeta function associated to an arbitrary ideal.
We derive transformation rules for test ideals and $F$-singularities under an arbitrary finite surjective morphism $π: Y \to X$ of normal varieties in prime characteristic $p > 0$. The main technique is to relate homomorphisms $F_{*} O_{X} \to O_{X}$, such as Frobenius splittings, to homomorphisms $F_{*} O_{Y} \to O_{Y}$. In the simplest cases, these rules mirror transformation rules for multiplier ideals in characteristic zero. As a corollary, we deduce sufficient conditions which imply that trace is surjective, i.e. $Tr_{Y/X}(π_{*}O_{Y}) = O_{X}$.
For a complex polynomial in two variables we study the morphism induced in homology by the embedding of an irregular fiber in a regular neighborhood of it. We give necessary and sufficient conditions for this morphism to be injective, surjective. Particularly this morphism is an isomorphism if and only if the corresponding irregular value is regular at infinity. We apply these results to the study of vanishing and invariant cycles.
The paper is a second step in the study of $\overline{M}_{0,n}$ started in arXiv:1006.0987 [math.AG]. We study fiber type morphisms of this moduli space via Kapranov's beautiful description. Our final goal is to understand if any dominant morphism $f: \overline{M}_{0,n} \to X$ with positive dimensional fibers factors through forgetful morphisms. We prove that this is true if either $n \leq 7$ or $\rm {dim} X \leq 2$ or the rational map induced on $P^{n-3}$ has linear general fibers. Along the way we give examples of forgetful morphisms whose fibers are connected curves of arbitrarily high posi…
We introduce the notion of combinatorial type of varieties $X$ which generalizes the concept of the dual complex of SNC divisors. It is a unique, up to homotopy, finite simplicial complex $Σ(X)$ which is functorial with respect to morphisms of varieties. Its cohomology $H^i(Σ(X),Q)$ for complex projective varieties coincide with weight zero part of the Deligne filtration $W_0(H^i(X,Q))$. The notion can be understood as a topological measure of the singularities of algebaric schemes of finite type. We also prove that any variety in characteristic zero admits the Hironaka desingularization with …
Fiber cones of 0-dimensional ideals with almost minimal multiplicity in Cohen-Macaulay local rings are studied. Ratliff-Rush closure of filtration of ideals with respect to another ideal is introduced. This is used to find a bound on the reduction number with respect to an ideal. Rossi's bound on reduction number in terms of Hilbert coefficients is obtained as a consequence. Sufficient conditions are provided for the fiber cone of 0-dimensional ideals to have almost maximal depth. Hilbert series of such fiber cones are also computed.
Our object of study is a rational map Psi from projective s-1 space to projective n-1 space defined by homogeneous forms g1,...,gn, of the same degree d, in the homogeneous coordinate ring R=k[x1,...,xs] of projective s-1 space. Our goal is to relate properties of Psi, of the homogeneous coordinate ring A=k[g1,...,gn] of the variety parametrized by Psi, and of the Rees algebra R[It], the bihomogeneous coordinate ring of the graph of Psi. For a regular map Psi, for instance, we prove that R[It] satisfies Serre's condition R_i, for some positive i, if and only if A satisfies R_{i-1} and Psi is b…
Let (R, m) be a Cohen-Macaulay local ring and I be an m-primary ideal. We introduce ideals of almost minimal mixed multiplicty which are analogues of ideals studied by J. Sally. The main theorem describes the Hilbert series of fiber cones of these ideals.
Let $f: (X,D) \to B$ be a stably family with log canonical general fiber. We prove that, after a birational modification of the base $\tilde{B} \to B$, there is a morphism from a high fibered power of the family to a pair of log general type. If in addition the general fiber is openly canonical, then there is a morphism from a high fibered power of the original family to a pair openly of log general type.
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.
A separable, proper morphism of varieties with geometrically connected fibers induces a homotopy exact sequence relating the étale fundamental groups of source, target and fiber. Extending work of dos Santos, we prove the existence of an analogous homotopy exact sequence for fundamental group schemes classifying regular singular stratified bundles, under the additional assumption that the morphism in question can be (partially) compactified to a log smooth morphism.
Analytic and Algebraic Oka-1 Approximation for Smooth Projective Morphisms with Rationally Connected Fibers
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学重点实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
We consider a rational map $ϕ: \mathbb{P}_k^{m} \dashrightarrow \mathbb{P}_k^n$ that is a parameterization of an $m$-dimensional variety. Our main goal is to study the $(m-1)$-dimensional fibers of $ϕ$ in relation to the $m$-th local cohomology modules of the Rees algebra of its base ideal.
We construct a new motivic integration morphism, the so-call bounded integral, that interpolates both the integration morphisms with and without volume forms of Hrushovski and Kazhdan. This is done within the framework of model theory of algebraically closed valued fields of equicharacteristic zero. As an application, we recover and extend some results of Hrushovski and Loeser about the motivic Milnor fiber.
In characteristic zero, we construct relative principalization of ideals for logarithmically regular morphisms of logarithmic schemes, and use it to construct logarithmically regular desingularization of morphisms. These constructions are relatively canonical and even functorial with respect to logarithmically regular morphisms and arbitrary base changes. Relative canonicity means, that the principalization requires a fine enough non-canonical modification of the base, and once it is chosen the process is canonical. As a consequence we deduce the semistable reduction theorem over arbitrary val…
Let $R$ be a polynomial ring and $I \subset R$ be a perfect ideal of height two minimally generated by forms of the same degree. We provide a formula for the multiplicity of the saturated special fiber ring of $I$. Interestingly, this formula is equal to an elementary symmetric polynomial in terms of the degrees of the syzygies of $I$. Applying ideas introduced in arXiv:1805.05180, we obtain the value of the j-multiplicity of $I$ and an effective method for determining the degree and birationality of rational maps defined by homogeneous generators of $I$.
The purpose of this paper is to establish injectivity theorems for higher direct image sheaves of canonical bundles twisted by pseudo-effective line bundles and multiplier ideal sheaves. As applications, we generalize Koll'ar's torsion freeness and Grauert-Riemenschneider's vanishing theorem. Moreover, we obtain a relative vanishing theorem of Kawamata-Viehweg-Nadel type and an extension theorem for holomorphic sections from fibers of morphisms to the ambient space. Our approach is based on transcendental methods and works for K"ahler morphisms and singular hermitian metrics with non-algebraic…
Let $R$ be a polynomial ring over a field and $I \subset R$ be a Gorenstein ideal of height three that is minimally generated by homogeneous polynomials of the same degree. We compute the multiplicity of the saturated special fiber ring of $I$. The obtained formula depends only on the number of variables of $R$, the minimal number of generators of $I$, and the degree of the syzygies of $I$. Applying results from arXiv:1805.05180, we get a formula for the $j$-multiplicity of $I$ and an effective method to study a rational map determined by a minimal set of generators of $I$.
The geometric Frobenius morphism on smooth varieties is an fppf-fiber bundle. We study representations of the structure group scheme. In particular, we describe irreducible representations and compute its Grothendieck ring of finite dimensional representations.
We give sufficient conditions to ensure that the ideal $Φ(\mathcal E)$ of $\mathcal E$-phantom maps in a locally $λ$-presentable exact category $(\mathcal{A}, \mathcal{E})$ is (special) (pre)covering ideal, where $\mathcal E$ is an exact substructure of $(\mathcal{A}, \mathcal{E})$. As a byproduct, we infer the existence of various covering ideals in categories of sheaves which have a meaningful geometrical motivation. In particular we deal with a Zariski-local notion of phantom maps in categories of sheaves. We would like to point out that our approach is necessarily different from [FGHT13], …
Hitchin in [Duke Math. J. 54 (1), 91-114 (1987)] introduced a proper morphism from the moduli space of stable $G$-Higgs bundles ($G=\mathrm{GL}(n,\mathbb{C}),\mathrm{Sp}(2m,\mathbb{C})$ and $\mathrm{SO}(n,\mathbb{C})$) over a curve to a vector space of invariant polynomials and he described the generic fibers of that morphism. In this paper, we first describe the generic Hitchin fibers for the moduli space of stable parabolic projective symplectic/orthogonal Higgs bundles without fixing the determinant. We also describe the generic fibers when the determinant is trivial.
The toric fiber product is an operation that combines two ideals that are homogeneous with respect to a grading by an affine monoid. The Segre product is a related construction that combines two multigraded rings. The quotient ring by a toric fiber product of two ideals is a subring of the Segre product, but in general this inclusion is strict. We contrast the two constructions and show that any Segre product can be presented as a toric fiber product without changing the involved quotient rings. This allows to apply previous results about toric fiber products to the study of Segre products. We…
Given a complete nonsingular algebraic variety $X$ and a divisor $D$ with normal crossings, we say that $X$ is log homogeneous with boundary $D$ if the logarithmic tangent bundle $T_X(- \log D)$ is generated by its global sections. We then show that the Albanese morphism $α$ is a fibration with fibers being spherical (in particular, rational) varieties. It follows that all irreducible components of $D$ are nonsingular, and any partial intersection of them is irreducible. Also, the image of $X$ under the morphism $σ$ associated with $- K_X - D$ is a spherical variety, and the irreducible compon…
Given an ideal $\mathcal I$ on a variety $X$ with toroidal singularities, we produce a modification $X' \to X$, functorial for toroidal morphisms, making the ideal monomial on a toroidal stack $X'$. We do this by adapting the methods of [Wło05], discarding steps which become redundant. We deduce functorial resolution of singularities for varieties with logarithmic structures. This is the first step in our program to apply logarithmic desingularization to a morphism $Z \to B$, aiming to prove functorial semistable reduction theorems.
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 prove a "Generic Equivalence Theorem which says that two affine morphisms $p: S \to Y$ and $q: T \to Y$ of varieties with isomorphic (closed) fibers become isomorphic under a dominant etale base change $ϕ: U \to Y$. A special case is the following result. Call a morphism $ϕ: X \to Y$ a "fibration with fiber $F$" if $ϕ$ is flat and all fibers are (reduced and) isomorphic to $F$. Then an affine fibration with fiber $F$ admits an etale dominant morphism $μ: U \to Y$ such that the pull-back is a trivial fiber bundle: $U\times_Y X \simeq U\times F$. As an application we give short proofs of the …
Let $X$ and $S$ be complex spaces with $X$ countable at infinity and $S$ reduced locally pure dimensional. Let $π:X\to S$ be an universally-$n$-equidimensional morphism (i.e open with constant pure $n$-dimensional fibers). If there is a cycle $\goth{X}$ of $X\times S$ such that, his support coincide fiberwise set-theorically with the fibers of $π$ and endowed this with a good multiplicities in such a way that $(π^{-1}(s))_{s\in S}$ becomes a local analytic (resp. continuous) family of cycles in the sense of [B.M], $π$ is called analytically(resp. continuously) geometrically flat according to t…
Let $K$ be a field of characteristic zero, $X$ and $Y$ be smooth $K$-varieties, and let $V$ be a finite dimensional $K$-vector space. For two algebraic morphisms $φ:X\rightarrow V$ and $ψ:Y\rightarrow V$ we define a convolution operation, $φ*ψ:X\times Y\to V$, by $φ*ψ(x,y)=φ(x)+ψ(y)$. We then study the singularity properties of the resulting morphism, and show that as in the case of convolution in analysis, it has improved smoothness properties. Explicitly, we show that for any morphism $φ:X\rightarrow V$ which is dominant when restricted to each irreducible component of $X$, there exists $N\i…
In this paper we study finite morphisms between irreducible projective varieties in terms of the morphisms they induce between the respective analytifications. The background for the principal result is as follows. Let $V'$ and $V$ be irreducible, projective varieties over an algebraically closed, non- archimedean valued field $k$ and $ϕ$ be a finite morphism $ϕ: V' \to V$. Let $x \in V^{an}(L)$, where $L/k$ is an algebraically closed complete non-archimedean valued field extension. We associate canonically to $x$ an $L$-point of the space $(V \times_k L)^{an}$ which lies on the fiber over $x$…
In this paper we study fiber type morphisms between moduli spaces of pointed rational curves. Via Kapranov's description we are able to prove that the only such morphisms are forgetful maps. This allow us to show that the Automorphism group of $\overline{M}_{0,n}$ is the permutation group on $n$ elements as soon as $n\geq 5$.
Let $G$ be a split reductive group over a field $k$ of arbitrary characteristic, chosen suitably. Let $X\to S$ be a smooth projective morphism of locally noetherian $k$-schemes, with geometrically connected fibers. We show that for each Harder-Narasimhan type $τ$ for principal $G$-bundles, all pairs consisting of a principal $G$-bundle on a fiber of $X\to S$ together with a given canonical reduction of HN-type $τ$ form an algebraic stack $Bun_{X/S}^τ(G)$ over $S$. The forgetful $1$-morphism $Bun_{X/S}^τ(G) \to Bun_{X/S}(G)$ to the algebraic stack of all principal $G$-bundles on fibers of $X\to…
We study the fibers of Mirkovic-Vilonen convolution morphisms. We prove their equidimensionality when all the coweights in question are minuscule, and some related statements. We give applications to saturation problems for structure constants of Hecke and representation rings. An erratum has been added to correct insufficiencies in the proofs of Lemma 9.4 and Corollary 9.5 as they appeared in the published version.
Given a smooth projective 3-fold Y, with $H^{3,0}(Y)=0$, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing 1-cycles in Y to the intermediate Jacobian J(Y). We study in this paper the existence of families of 1-cycles in Y for which this induced morphism is surjective with rationally connected general fiber, and various applications of this property. When Y itself is rationally connected, we relate this property to the existence of an integral homological decomposition of the diagonal. We also study this property for cubic threefolds, completing the work of Iliev-Ma…
Relative $\mathbb{A}^1$-Contractibility of Smooth Schemes
CNRS, Université de Poitiers, LMA（法国国家科学研究中心，普瓦捷大学，数学实验室）；Université Bourgogne Europe, CNRS, IMB UMR 5584（勃艮第欧洲大学，法国国家科学研究中心，IMB UMR 5584）；Department of Mathematics “F. Enriques”, University of Milan（米兰大学，“F.恩里克斯”数学系）
Given a smooth projective 3-fold Y, with $H^{3,0}(Y)=0$, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing 1-cycles in Y to the intermediate Jacobian J(Y). We study in this paper the existence of families of 1-cycles in Y for which this induced morphism is surjective with rationally connected general fiber, and various applications of this property. When Y itself is rationally connected, we relate this property to the existence of an integral homological decomposition of the diagonal. We also study this property for cubic threefolds, completing the work of Iliev-Ma…
Let $f:X\to Y$ be a smooth morphism of complex analytic manifolds and let $F$ be an $\mathbb{R}$-constructible complex on $Y$. Let $\cal{M}$ be a coherent $\shd_X$-module. We prove that the microsupport of the solution complex of $\shm$ in the tempered holomorphic functions $t \shh \text{om} (f^{-1} F, \sho_X)$, is contained in the 1-characteristic variety of $\cal{M}$ associated to $f$, and that the microsupport of the solution complex in the tempered microfunctions $tμhom(f^{-1}F, \sho_X)$ is contained in the 1-microcharacteristic variety of the microlocalized of $\shm$ along $T^*Y\times_Y X…
Let $K$ be a non-Archimedean valued field with valuation ring $R$. Let $C_η$ be a $K$-curve with compact type reduction, so its Jacobian $J_η$ extends to an abelian $R$-scheme $J$. We prove that an Abel-Jacobi map $ι\colon C_η\to J_η$ extends to a morphism $C\to J$, where $C$ is a compact-type $R$-model of $J$, and we show this is a closed immersion when the special fiber of $C$ has no rational components. To do so, we apply a rigid-analytic "fiberwise" criterion for a finite morphism to extend to integral models, and geometric results of Bosch and Lütkebohmert on the analytic structure of $J_…
