Luis Fuentes Garcia
Given an etale quotient q:X->Y of smooth projective varieties we relate the simple Seshadri constant of a line bundle M on Y with the multiple Seshadri constant of q*M in the points of the fiber. We apply this method to compute the Seshadri constant of polarized abelian surfaces in the points of a finite subgroup.
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We prove that the Seshadri constant of a polarized abelian variety is equal to the Seshadri constant of its abelian subvariety if the Seshadri constant is relatively small with respect to its degree, or it contains an abelian divisor which has sufficiently small degree. As an application of these results, we show that the Seshadri constant of a polarized abelian surface is computed by the unique Seshadri curve if the Seshadri constant is small enough. As another application, we compute the Seshadri constants of some non-simple polarized abelian threefolds which contain principally polarized ab…
So far, Seshadri constants on abelian surfaces are completely understood only in the cases of Picard number one and on principally polarized abelian surfaces with real multiplication. Beyond that, there are partial results for products of elliptic curves. In this paper, we show how to compute the Seshadri constant of any nef line bundle on any abelian surface over the complex numbers. We develop an effective algorithm depending only on the basis of the Néron-Severi group to compute not only the Seshadri constants but also the numerical data of their Seshadri curves. Access to the Seshadri curv…
In this note we contribute to the study of Seshadri constants on abelian and bielliptic surfaces. We specifically focus on bounds that hold on all such surfaces, depending only on the self-intersection of the ample line bundle under consideration. Our result improves previous bounds and it provides rational numbers as bounds, which are potential Seshadri constants.
Seshadri constants of indecomposable polarized abelian varieties
本文研究不可分解极化阿贝尔簇的Seshadri常数下界猜想，受Debarre思想启发，并借助Newton-Okounkov体的凸几何方法，推广作者先前关于阿贝尔三维簇合冲的工作。
Consider a polarized abelian variety $(A,L)$ over the field of complex numbers. Following Demailly, one can associate to $(A,L)$ a real number $ε(A,L)$, its {\em Seshadri constant}, which in effect measures how much of the positivity of $L$ can be concentrated at any given point of $A$. There has been considerable recent interest in finding bounds on the Seshadri constants of abelian varieties and on smooth projective varieties in general. In the present paper we first generalize an approach of Buser and Sarnak to give a lower bound on the minimal period length of $(A,L)$ in terms of the type …
In this paper we consider the question of when Seshadri constants on abelian surfaces are integers. Our first result concerns self-products $E\times E$ of elliptic curves: If $E$ has complex multiplication in $\Z[i]$ or in $\Z[\frac12(1+i\sqrt3)]$ or if $E$ has no complex multiplication at all, then it is known that for every ample line bundle $L$ on $E\times E$, the Seshadri constant $\eps(L)$ is an integer. We show that, contrary to what one might expect, these are in fact the only elliptic curves for which this integrality statement holds. Our second result answers the question how -- on an…
Seshadri constants on principally polarized abelian surfaces with real multiplication
本文研究 Picard 数为 2 的具有实乘的主极化阿贝尔曲面上的 Seshadri 函数，证明其局部呈现 Cantor 函数式的复杂分段线性结构，同时整体上具有无限自同构群不变性。
Jets-separation thresholds, Seshadri constants and higher Gauss-Wahl maps on abelian varieties
本文在极化阿贝尔簇上定义闭子概形的消失阈值与射流分离阈值，并利用Fourier-Mukai方法建立其与高阶Gauss-Wahl映射满射性及Seshadri常数之间的联系，给出满射性判别准则。
Seshadri constants and hyperelliptic curves on abelian varieties
研究极化阿贝尔簇上光滑曲线的Seshadri常数与曲线度数之比的上界，通过曲线的高阶Gauß-Wahl映射的非满射性定义内在不变量，并证明超椭圆曲线的Castelnuovo型不等式及其等号条件。
The purpose of this note is to point out an elementary but somewhat surprising connection between the work of Buser and Sarnak on lengths of periods of abelian varieties and the Seshadri constants measuring the local positivity of theta divisors. The link is established via symplectic blowing up, in the spirit of McDuff and Polterovich. As an application of the main inequality, we get a simple new proof of a statement of Buser-Sarnak type to the effect that the Jacobian of a curve has a period of unusually small length. We also deduce a lower bound on the Seshadri constant of a very general p.…
Seshadri constants on algebraic surfaces
本文研究代数曲面上的 Seshadri 常数，证明了若干显式界，并对 Picard 数为 1 的阿贝尔曲面给出了可显式计算 Seshadri 常数及相应不可约曲线的完整结果，揭示了其数论特征。
Let $X$ be a smooth projective complex variety of maximal Albanese dimension, and let $L \to X$ be a big line bundle. We prove that the moving Seshadri constants of the pull-backs of $L$ to suitable finite abelian étale covers of $X$ are arbitrarily large. As an application, given any integer $k\geq 1$, there exists an abelian étale cover $p\colon X' \to X$ such that the adjoint system $\big|K_{X'} + p^*L \big|$ separates $k$-jets away from the augmented base locus of $p^*L$, and the exceptional locus of the pull-back of the Albanese map of $X$ under $p$.
Given a smooth complex projective variety $X$ and an ample line bundle $L$ on $X$. Fix a point $x\in X$. We consider the question, are there conditions which guarantee the maxima of the Seshadri constant of $L$ at $x$, i.e $\eps(L,x)=\root n \of {L^n}$? We give a partial answer for surfaces and find examples where the answer to our question is negative. If $(X,Θ)$ is a general principal polarized abelian surface, then $\eps(Θ,x)={4/3}<\sqrt{2}=\sqrt{Θ^2}$ for all $x\in X$.
Based on the theory of an infinitesimal Newton-Okounkov body, we extend the results of Lazarsfeld-Pareschi-Popa on abelian surfaces. Moreover, we show that the higher syzygies of $(X,L)$ are completely determined by its Seshadri constant when $L^{2}$ is large. As an application, we improve the existing lower bound of $(L^{2})$ for higher syzygies of a polarized abelian surface $(X,L)$.
On abelian extensions of finite abelian subgroups of Cremona groups
本文研究Cremona群有限阿贝尔子群的扩张性质，证明在维数至多4时扩张与直积一致（除一例外），从而对有限阿贝尔群情形下忠实作用提升至终端Fano簇的问题给出肯定回答。
For a semi-stable abelian variety A_K over a complete discrete valuation field K, we show that every finite subgroup scheme of A_K extends to a log finite flat group scheme over the valuation ring of K endowed with the canonical log structure. To achieve this, we first prove that every weak log abelian variety over an fs log scheme with its underlying scheme locally noetherian, is a sheaf for the Kummer flat topology, which answers a question of Chikara Nakayama. We also give several equivalent conditions defining isogenies of log abelian varieties.
Let G be a finite subgroup of SL(n,C), then the quotient C^n/G has a Gorenstein canonical singularity. Bridgeland-King-Reid proved that the G-Hilbert scheme Hilb^G(C^3) gives a crepant resolution of the quotient C^3/G for any finite subgroup G of SL(3,C). However, in dimension 4, very few crepant resolutions are known. In this paper, we will show several examples of crepant resolutions in dimension 4 and show examples in which Hilb^G(C^4) is blow-up of certain crepant resolutions for C^4/G, or Hilb^G(C^4) has singularity.
For any finite subgroup G in SL3(C), work of Bridgeland-King-Reid constructs an equivalence between the G-equivariant derived category of C^3 and the derived category of the crepant resolution Y = G-Hilb(C^3) of C^3/G. When G is abelian we show that this equivalence gives a natural correspondence between irreducible representations of G and certain sheaves on exceptional subvarieties of Y, thereby extending the McKay correspondence from two to three dimensions. This categorifies Reid's recipe and extends earlier work from [CL09] and [Log10] which dealt only with the case when C^3/G has one iso…
Finite abelian subgroups in the groups of birational and bimeromorphic selfmaps
该研究针对复射影簇的双有理自同构群中含任意大阶的固定秩有限阿贝尔子群的问题，证明秩不超过簇维数的两倍，等号成立等价于簇双有理等价于阿贝尔簇，并将结果推广到紧Kähler空间的双亚纯自同构群。
The problem of enumeration of conjugacy classes of finite abelian subgroups of the mapping class group $\mathcal{M}_σ$ of a compact closed surface $X$ of genus $σ$ is considered. A complete method of enumeration is achieved for finite elementary abelian subgroups and steps are taken toward enumeration of finite abelian subgroups.
Given an abelian variety over a field with a discrete valuation, Grothendieck defined a certain open normal subgroup of the absolute inertia group. This subgroup encodes information on the extensions over which the abelian variety acquires semistable reduction. We study this subgroup, and use it to obtain information on the extensions over which the abelian variety acquires semistable reduction.
For any $p>2$ we give an example of big action $(X,G)$ with non abelian derived subgroup. It is obtained as a covering of a curve related to the Ree curve.
On abelian points of varieties intersecting subgroups in a torus
在自然条件下，研究了闭不可约子簇X与托尔中连通代数子群的并集的非异常子集上的阿贝尔点集是有限的，并推广了Ostafe、Sha、Shparlinski和Zannier(2017)的结果，同时在非连通代数子群情况下推广了其结构定理，并在曲线和算术动力系统中得到了相关结果。
In this paper, we introduce an algebro-geometric formulation for Faltings' theorem on diophantine approximation on abelian varieties using an improvement of Faltings-Wustholz observation over number fields. In fact, we prove that, for any geometrically irreducible sub-variety E of an abelian variety A and any finitely generated subgroup F of A(C) we have an estimate of the form d_v(E;x) >cH(x)^d for for some constant c where d_v(E;x) denotes the distance of a point x in F outside E and v is a place of K. This was proved before, only for F being the set of rational points of A over a number fie…
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