Linear projections and successive minima
英文摘要：
Using linear projections one gets new inequalities for the successive minima of the lattice of sections of an hermitian line bundle on an arithmetic surface.
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Erratum : Linear projections and successive minima
Given a euclidean lattice and a curve in the corresponding projective space, our goal is to relate the height of the curve to the successive minima of the lattice. The proof of such a statement in http://hal.archves-ouvertes.vf/hal-00270564 was incorrect. We prove another result instead.
math.AG
Lower bounds for Seshadri constants via successive minima of line bundles
Given a nef and big line bundle $L$ on a projective variety $X$ of dimension $d \geq 2$, we prove that the Seshadri constant of $L$ at a very general point is larger than $(d+1)^{\frac{1}{d}-1}$. This slightly improves the lower bound $1/d$ established by Ein, Küchle and Lazarsfeld. The proof relies on the concept of successive minima for line bundles recently introduced by Ambro and Ito.
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Successive minima of line bundles
We introduce and study the successive minima of line bundles on proper algebraic varieties. The first (resp. last) minima are the width (resp. Seshadri constant) of the line bundle at very general points. The volume of the line bundle is equivalent to the product of the successive minima. For line bundles on toric varieties, the successive minima are equivalent to the (reciprocal of) usual successive minima of the difference of the moment polytope.
math.AGmath.CO
Successive minima of toric height functions
Given a toric metrized R-divisor on a toric variety over a global field, we give a formula for the essential minimum of the associated height function. Under suitable positivity conditions, we also give formulae for all the successive minima. We apply these results to the study, in the toric setting, of the relation between the successive minima and other arithmetic invariants like the height and the arithmetic volume. We also apply our formulae to compute the successive minima for several families of examples, including weighted projective spaces, toric bundles and translates of subtori.
math.AGmath.NT
Nef锥与连续极小值：一个例子
Nef cone and successive minima: an example
本文研究了平滑射影曲线与其雅可比变种积空间的nef锥和伪有效锥，并展示了张定理在该情况下不成立。
math.AGmath.NT
Secant varieties and successive minima
Given an arithmetic surface and a positive hermitian line bundle over it, we bound the successive minima of the lattice of global sections of this line bundle. Our method combines a result of C.Voisin on secant varieties of projective curves with previous work by the author on the arithmetic analog of the Kodaira vanishing theorem. The paper also includes a result of A.Granville on the divisibility properties of binomial coefficients in a given line of Pascal's triangle.
math.AGmath.NT
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数域中序的逐次极小值的界与曲线的滚动不变量
Bounds on Successive Minima of Orders in Number Fields and Scrollar Invariants of Curves
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Infinitesimal successive minima, partial jets and convex geometry
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Minimal set of binomial generators for certain Veronese 3-fold projections
The goal of this paper is to explicitly describe a minimal binomial generating set of a class of lattice ideals, namely the ideal of certain Veronese $3$-fold projections. More precisely, for any integer $d\ge 4$ and any $d$-th root $e$ of 1 we denote by $X_d$ the toric variety defined as the image of the morphism $φ_{T_d}:\mathbb{P}^3 \longrightarrow \mathbb{P}^{μ(T_d)-1}$ where $T_d$ are all monomials of degree $d$ in $k[x,y,z,t]$ invariant under the action of the diagonal matrix $M(1,e,e^2,e^3).$ In this work, we describe a $\mathbb{Z}$-basis of the lattice $L_{η}$ associated to $I(X_d)$ as…
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