José Carlos Sierra
A classification and a detailed geometric description are given for smooth $n$-dimensional subvarieties $X\subset{\mathbb P}^{2n-1}$ containing a family of effective divisors each of them spanning a linear ${\mathbb P}^n$ of ${\mathbb P}^{2n-1}$. Some results on multisecant lines to threefolds $X\subset{\mathbb P}^5$ follow, as a byproduct.
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In this paper we consider ideal sheaves associated to the singular loci of a divisor in a linear system $|L|$ of an ample line bundle on a complex abelian variety. We prove an effective result on their (continuous) global generation, after suitable twists by powers of $L$. Moreover we show that similar results hold for subvarieties of a complex abelian variety.
We introduce a notion of ampleness for subschemes of higher codimension using the theory of q-ample line bundles. We also investigate certain geometric properties satisfied by ample subvarieties, e.g. the Lefschetz hyperplane theorems and numerical positivity. Using these properties, we also construct a counterexample to the converse of the Andreotti-Grauert vanishing theorem.
Rational curves on Hilbert schemes of points on $K3$ surfaces and generalised Kummer manifolds are constructed by using Brill-Noether theory on nodal curves on the underlying surface. It turns out that all wall divisors can be obtained, up to isometry, as dual divisors to such rational curves. The locus covered by the rational curves is then described, thus exhibiting algebraically coisotropic subvarieties that deform to general small deformations of the manifold. This provides strong evidence for a conjecture by Voisin concerning the Chow ring of irreducible holomorphic symplectic manifolds. …
An Okounkov body is a convex subset in Euclidean space associated to a big divisor on a smooth projective variety with respect to an admissible flag. In this paper, we introduce two convex bodies associated to pseudoeffective divisors, called the valuative Okounkov bodies and the limiting Okounkov bodies, and show that these convex bodies reflect the asymptotic properties of pseudoeffective divisors as in the case with big divisors. Our results extend the works of Lazarsfeld-Mustaţă and Kaveh-Khovanskii. For this purpose, we define and study special subvarieties, called the Nakayama subvarieti…
We show that for an irreducible subvariety Y of an abelian variety X the Gauss mapping, from the conormal bundle of $Y$ to the dual of the tangent space of $X$ at the origin, is not dominant if and only if Y is degenerate in the sense that there exists a nontrivial abelian subvariety A of X such that A+Y=Y holds
This paper proves a finiteness result for families of integral points on a semiabelian variety minus a divisor, generalizing the corresponding result of Faltings for abelian varieties. Combined with the main theorem of the first part of this paper, this gives a finiteness statement for integral points on a closed subvariety of a semiabelian variety, minus a divisor. In addition, the last two sections generalize some standard results on closed subvarieties of semiabelian varieties to the context of closed subvarieties minus divisors.
We consider generic degenerate subvarieties $X_i\subset\mathbb{P}^n$. We determine an integer $N$, depending on the varieties, and for $n\geq N$ we compute dimension and degree formulas for the Hadamard product of the varieties $X_i$. Moreover, if the varieties $X_i$ are smooth, their Hadamard product is smooth too. For $n<N$, if the $X_i$ are generically $d_i$-parameterized, the dimension and degree formulas still hold. However, the Hadamard product can be singular and we give a lower bound for the dimension of the singular locus.
We show that a pseudoeffective R-divisor has numerical dimension 0 if it is numerically trivial on a subvariety with ample normal bundle. This implies that the cycle class of a curve with ample normal bundle is big, giving an affirmative answer to a question of Peternell. We also give other positivity properties such subvarieties.
We explore some of the interplay between Brill-Noether subvarieties of the moduli space SU_C(2,K) of rank 2 bundles with canonical determinant on a smooth projective curve and 2θdivisors, via the inclusion of the moduli space into |2θ|, singular along the Kummer variety. In particular we show that the moduli space contains all the trisecants of the Kummer and deduce that there are quadrisecant lines only if the curve is hyperelliptic; we show that for generic curves of genus <6, though no higher, bundles with >2 sections are cut out by Γ_00; and that for genus 4 this locus is precisely the Don…
We show that if $X$ is a smooth uniruled projective variety and $L$ a big and semiample $\mathbb{Q}$-divisor on $X$, then there exists a proper closed subset $W\subset X$ such that every subvariety $Y$ satisfying $a(Y,L)> a(X,L)$ is contained in $W$.
For a compact Kahler manifold X and a strongly primitive automorphism g of positive entropy, it is shown that X has at most rank NS(X) of g-periodic prime divisors B_i (i.e., g^s(B_i) = B_i for some s > 0). When X is a projective threefold, every prime divisor containing infinitely many g-periodic curves, is shown to be g-periodic itself (a result in the spirit of the Dynamic Manin-Mumford conjecture).
A log resolution for the theta divisor of a hyperelliptic curve
该研究针对光滑超椭圆曲线θ除子的奇点消解问题，提出通过迭代吹张Brill-Noether子簇得到自然显式的嵌入消解，并证明超椭圆雅可比簇的Brill-Noether分层为Whitney分层，为代数几何中相关奇点研究提供了新的显式构造结果。
In this paper, I prove a very general extension theorem for log pluricanonical systems. The main application of this extension theorem is (together with Kawamata's subadjunction theorem) to give an optimal subadjunction theorem which relates the positivities of canonical bundle of the ambient projective manifold and that of the (maximal) center of log canonical singularities. This is an extension of the corresponding result in my previous work where I dealt with log pluricanonical systems of general type. This subadjunction theorem indicates an approach to solve the abundance conjecture for ca…
Finiteness and rigidity of optimal destabilizing subvarieties for the J-equation
该研究证明J方程无光滑解时，紧致Kähler流形上最优去稳定化曲线与除子有限，三维流形上所有此类子簇无条件有限，还刻画其轨迹并证明最优闭链的刚性准则。
Inside the symmetric product of a very general curve, we consider the codimension-one subvarieties of symmetric tuples of points imposing exceptional secant conditions on linear series on the curve of fixed degree and dimension. We compute the classes of such divisors, and thus obtain improved bounds for the slope of the cone of effective divisor classes on symmetric products of a very general curve. By letting the moduli of the curve vary, we study more generally the classes of the related divisors inside the moduli space of stable pointed curves.
Classification of Smooth Minimal Kähler Fourfolds Without Effective Divisors and Surfaces
研究紧致凯勒四维流形\(X\)，其典范丛伪有效且无余维数一二的子簇，证明\(K_X\)是挠线丛，借助Beauville - Bogomolov分解定理得出\(X\)是复环面商或不可约全纯辛流形。
We study the problem of classifying the irreducible projective varieties $X$ of dimension $n\ge 2$ in $\Bbb P^N$ which contain an algebraic family $\Cal F$ of dimension $h+1$ ($h<n$) of subvarieties $Y$ of dimension $n-h$, each one contained in a $\Bbb P^{N-h-1}$. We prove that one of the following happens: (i) there exists an integer $r$, $r<N-n$ such that $X$ is contained in a variety $V_r$ of dimension at most $N-r$ containing a family of dimension $h+1$ of subvarieties of dimension $N-h-r$, each one contained in a linear space of dimension $N-h-1$; (ii) The degree of $Y$ is bounded by a fu…
Let $(X,o)$ be a 3-dimensional terminal singularity of type $cD$ or $cE$ defined in $\mathbb{C}^4$ by an equation non-degenerate with respect to its Newton diagram. We show that there is not more than 1 non-rational divisor $E$ over $(X,o)$ with discrepancy $a(E,X)=1$. We also describe all blowups $σ$ of $(X,o)$ such that $E=\Exc(σ)$ is non-rational and $a(E,X)=1$.
We work over an algebraically closed field of arbitrary characteristic. Ellingsrud-Peskine proved that smooth surfaces in P^4 are subject to strong limitations. Their whole argument is derived from the fact that the sectional genus of surfaces of degree d lying on a hypersurface of degree s varies in an interval of length \frac{d(s-1)^2}{2s}. The aim of the present paper is to show that for smooth codimension two subvarieties of P^n, n\geq 5, one can get a similar result with an interval whose length depends only on s. The main point is Lemma 1.1 whose proof is a direct application of the posi…
Fix integers $a\geq 1$, $b$ and $c$. We prove that for certain projective varieties $V\subset{\bold P}^r$ (e.g. certain possibly singular complete intersections), there are only finitely many components of the Hilbert scheme parametrizing irreducible, smooth, projective, low codimensional subvarieties $X$ of $V$ such that $$ h^0(X,\Cal O_X(aK_X-bH_X)) \leq λd^{ε_1}+c(\sum_{1\leq h < ε_2}p_g(X^{(h)})), $$ where $d$, $K_X$ and $H_X$ denote the degree, the canonical divisor and the general hyperplane section of $X$, $p_g(X^{(h)})$ denotes the geometric genus of the general linear section of $X$ o…
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.
Let $(M,I,J,K)$ be a hyperkahler manifold, and $Z\subset (M,I)$ a complex subvariety in $(M,I)$. We say that $Z$ is trianalytic if it is complex analytic with respect to $J$ and $K$, and absolutely trianalytic if it is trianalytic with respect to any hyperkähler triple of complex structures $(M,I,J',K')$ containing $I$. For a generic complex structure $I$ on $M$, all complex subvarieties of $(M,I)$ are absolutely trianalytic. It is known that a normalization $Z'$ of a trianalytic subvariety is smooth; we prove that $b_2(Z')$ is no smaller than $b_2(M)$ when $M$ has maximal holonomy (that is, $…
We give two characterizations of hyperquadrics: one as non-degenerate smooth projective varieties swept out by large dimensional quadric subvarieties passing through a point; the other as $LQEL$-manifolds with large secant defects.
Let $X$ be a normal projective variety defined over an algebraically closed field and let $Z$ be a subvariety. Let $D$ be an $\mathbb R$-Cartier $\mathbb R$-divisor on $X$. Given an expression $(\ast) \ D \sim_{\mathbb R} t_1 H_1 + \ldots + t_s H_s$ with $t_i \in \mathbb R$ and $H_i$ very ample, we define the $(\ast)$-restricted volume of $D$ to $Z$ and we show that it coincides with the usual restricted volume when $Z \not\subseteq \mathbf{B}_+(D)$. Then, using some recent results of Birkar (arXiv:1312.0239), we generalize to $\mathbb R$-divisors the two main results of arXiv:1305.4284 by Bou…
We prove that any compact Kähler 3-dimensional manifold which has no non-trivial complex subvarieties is a torus. This is a very special case of a general conjecture on the structure of 'simple manifolds', central in the bimeromorphic classification of compact Kähler manifolds. The proof follows from the Brunella pseudo-effectivity theorem, combined with fundamental results of Siu and of the second author on the Lelong numbers of closed positive (1,1)-currents, and with a version of the hard Lefschetz theorem for pseudo-effective line bundles, due to Takegoshi and Demailly-Peternell-Schneider.…
We generalize Nakamaye's description, via intersection theory, of the augmented base locus of a big and nef divisor on a normal pair with log-canonical singularities or, more generally, on a normal variety with non-lc locus of dimension at most 1. We also generalize Ein-Lazarsfeld-Mustata-Nakamaye-Popa's description, in terms of valuations, of the subvarieties of the restricted base locus of a big divisor on a normal pair with klt singularities.
In this paper a new geometric characterization of the $n$th symmetric product of a curve is given. Specifically, we assume that there exists a chain of smooth subvarieties $V_i$ of dimension $i$, such that $V_i$ is an ample divisor in $V_{i+1}$ and its intersection product with $V_1$ is one. That the Albanese dimension of $V_2$ is $2$ and the genus of $V_1$ is equal to the irregularity of the variety. We prove that in this case the variety is isomorphic to the symmetric product of a curve.
We study the rational Picard group of the projectivized moduli space of holomorphic n-differentials on complex genus g stable curves. We define (n - 1) natural classes in this Picard group that we call Prym-Tyurin classes. We express these classes as linear combinations of boundary divisors and the divisor of n-differentials with a double zero. We give two different proofs of this result, using two alternative approaches: an analytic approach that involves the Bergman tau function and its vanishing divisor and an algebro-geometric approach that involves cohomological computations on the univer…
Unimodal singularities and boundary divisors in the KSBA moduli of a class of Horikawa surfaces
本文研究一类Horikawa曲面典范模型模空间的KSBA紧化，构造了八个新的不可约边界除子，并探讨其与GIT紧化及退化曲面Hodge理论的关系。
Let $(X,o)$ be a germ of a 3-dimensional terminal singularity of index $m\geq 2$. If $(X,o)$ has type cAx/4, cD/3-3, cD/2-2, or cE/2, then assume that the standard equation of $X$ in $\mathbb{C}^4/\mathbb{Z}_m$ is non-degenerate with respect to its Newton diagram. Let $π\colon Y\to X$ be a resolution. We show that there are not more than 2 non-rational divisors $E_i$, $i=1,2$, on $Y$ such that $π(E_i)=o$ and discrepancy $a(E_i,X)\leq 1$. When such divisors exist, we describe them as exceptional divisors of certain blowups of $X$ and study their birational type.
A collection $S = \{D_1,\ldots, D_n\}$ of divisors in a smooth variety $X$ is an {\em arrangement} if intersections of all subsets of $S$ are smooth. We show that a double cover of $X$ ramified on an arrangement has a crepant resolution under additional hypotheses. Namely, we assume that all intersection components that change the canonical divisor when blown up satisfy are {\em splayed}, a property of the tangent spaces of the components first studied by Faber. This strengthens a result of Cynk and Hulek, which requires a stronger hypothesis on the intersection components. Further, we study t…
For conic bundles on a smooth variety (over a field of characteristic $\ne 2$) which degenerate into pairs of distinct lines over geometric points of a smooth divisor, we prove a theorem which relates the Brauer class of the non-degenerate conic on the complement of the divisor to the covering class (Kummer class) of the 2-sheeted cover of the divisor defined by the degenerate conic, via the Gysin homomorphism in etale cohomology. This theorem is the algebro-geometric analogue of a topological result proved earlier.
We give an explicit formula for the Hodge filtration on the $\mathscr{D}_X$-module $O_X(*Z)f^{1-α}$ associated to the effective $\mathbb{Q}$-divisor $D=α\cdot Z$, where $0<α\le1$ and $Z=(f=0)$ is an irreducible hypersurface defined by $f$, a weighted homogeneous polynomial with an isolated singularity at the origin. In particular this gives a formula for the Hodge ideals of $D$. We deduce a formula for the generating level of the Hodge filtration, as well as further properties of Hodge ideals in this setting. We also extend the main theorem to the case when $f$ is a germ of holomorphic functio…
Going one step further in Zak's classification of Scorza varieties with secant defect equal to one, we characterize the Veronese embedding of $¶^n$ given by the complete linear system of quadrics and its smooth projections from a point as the only smooth irreducible complex and non-degenerate projective subvarieties of $¶^N$ that can be projected isomorphically into $¶^{2n}$ when $N\geq\binom{n+2}{2}-2$.
We prove a decomposition formula of logarithmic Gromov-Witten invariants in a degeneration setting. A one-parameter log smooth family X->B with singular fibre over b_0 \in B yields a family M(X/B,β) -> B of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over b_0 in terms of rigid tropical curves. This generalizes one aspect of known results in the case that the fibre X_{b_0} is a normal crossings union of two divisors. We exhibit our formulas in explicit examples.
Let $D$ be a very general curve of degree $d=2\ell-ε$ in $\mathbb{P}^2$, with $ε\in \{0,1\}$. Let $Γ\subset \mathbb{P}^2$ be an integral curve of geometric genus $g$ and degree $m$, $Γ\neq D$, and let $ν: C\to Γ$ be the normalization. Let $δ$ be the degree of the \emph{reduction modulo 2} of the divisor $ν^*(D)$ of $C$. In this paper we prove the inequality $4g+δ\geqslant m(d-8+2ε)+5$. We compare this with similar inequalities due to Geng Xu and Xi Chen. Besides, we provide a brief account on genera of subvarieties in projective hypersurfaces.
For $k \in \mathbb{Z}_{>0}$, let $\mathcal{H}^{(k)}_{g,n}$ denote the vector bundle over $\mathfrak{M}_{g,n}$ whose every fiber consists of meromorphic $k$-differentials with poles of order at most $k-1$ on a fixed Riemman surface of genus $g$ with $n$ marked points (all the poles must be located at the marked points). The bundle $\mathcal{H}^{(k)}_{g,n}$ and its associated projective bundle $\mathbb{P}\mathcal{H}^{(k)}_{g,n}$ admit natural extensions, denoted by $\overline{\mathcal{H}}^{(k)}_{g,n}$ and $\mathbb{P}\overline{\mathcal{H}}^{(k)}_{g,n}$ respectively, to the Deligne-Mumford compact…
We introduce logarithmic Picard algebroids, a natural class of Lie algebroids adapted to a simple normal crossings divisor on a smooth projective variety. We show that such algebroids are classified by a subspace of the de Rham cohomology of the divisor complement determined by its mixed Hodge structure. We then solve the prequantization problem, showing that under the appropriate integrality condition, a log Picard algebroid is the Lie algebroid of symmetries of what is called a meromorphic line bundle, a generalization of the usual notion of line bundle in which the fibres degenerate along t…
This paper begins with a description of cohomological invariants of non-degenerate quadric bundles, in terms of the cohomology rings of the classifying spaces of the general orthogonal groups. Following this, the Main Theorem of the paper determines the behavior of these invariants under the Gysin boundary map, when a quadric bundle degenerates over a divisor.
Let $X \subseteq \mathbb{P}^r$ be a non-degenerate smooth projective variety of dimension $n$, codimension $e$, and degree $d$ defined over an algebraically closed field of characteristic zero. In this paper, we first show that $\text{reg} (\mathcal{O}_X) \leq d-e$, and classify the extremal and the next to extremal cases. Our result reduces the Eisenbud-Goto regularity conjecture for the smooth case to the problem finding a Castelnuovo-type bound for normality. It is worth noting that McCullough-Peeva recently constructed counterexamples to the regularity conjecture by showing that $\text{reg…
Campana's orbifold conjecture for numerically equivalent divisors
本文证明了数值平行除子条件下的Campana轨形猜想，证明了满足特定重数条件的轨形整曲线必然代数退化。
Rational Homotopy Type of Complements of Submanifold Arrangements
本文通过构造cdga模型，研究了子流形排列补集的有理同调类型，并推广了多个关于排列补集的已知结果。
Given a family $X$ of complex varieties degenerating over a punctured disc, one is interested in computing related invariants called the motivic nearby fiber and the refined limit mixed Hodge numbers, both of which contain information about the induced action of monodromy on the cohomology of a fiber of $X$. Our first main result is that the motivic nearby fiber of $X$ can be computed by first stratifying $X$ into locally closed subvarieties that are non-degenerate in the sense of Tevelev, and then applying an explicit formula on each piece of the stratification that involves tropical geometry…
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