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.
A variety of minimal degree is one of the basic objects in projective algebraic geometry and has been classified and characterized in many aspects. On the other hand, there are also minimal objects in the category of higher secant varieties, and their algebraic and geometric structures seem to share many similarities with those of varieties of minimal degree. We prove in this paper that higher secant varieties of minimal degree have determinantal presentation of two types, i.e. scroll type and Veronese type. Our result generalizes the del Pezzo-Bertini classification for varieties of minimal d…
This paper studies the dimension of secant varieties to Segre varieties. The problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry. An inductive procedure is built around the ideas of successive specializations of points and projections. This reduces the calculation of the dimension of the secant variety in a high dimensional case to a sequence of calculations of partial secant varieties in low dimensional cases. As applications of the technique: We give a complete classification of defective $t$-secant varieties to Segre varieties for t < 7. We general…
To complete the classification theory and the structure theory of varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2, a natural approach is to investigate simple projections of varieties of minimal degree. Let $\tilde X \subset {\mathbb P}^{r+1}_K$ be a variety of minimal degree and of codimension at least 2, and consider $X_p = π_p (\tilde X) \subset {\mathbb P}^r_K$ where $p \in {\mathbb P}^{r+1}_K \backslash \tilde X$. By \cite{B-Sche}, it turns out that the cohomological and local properties of …
In this paper we prove, using a refinement of Terracini's Lemma, a sharp lower bound for the degree of (higher) secant varieties to a given projective variety, which extends the well known lower bound for the degree of a variety in terms of its dimension and codimension in projective space. Moreover we study varieties for which the bound is attained proving some general properties related to tangential projections, e.g. these varieties are rational. In particular we completely classify surfaces (and curves) for which the bound is attained. It turns out that these surfaces enjoy some maximality…
We try to write (generic) elements of classical simple Lie algebras as sums of as few elements of the orbit C of long root vectors as possible. If the Lie algebra in question is sl_n or sp_n, then C consists of all rank 1 matrices in the Lie algebra, and an element of rank k is the sum of k elements of C. If, on the other hand, the Lie algebra is o_n, then C consists of all rank 2 matrices with square zero, and the situation is much more complicated. We explicitly describe, in this case as well, the closure of the set of all sums of k elements of C, also known as the (k-1)st secant variety of …
When present, the Cohen-Macaulay property can be useful for finding the minimal defining equations of an algebraic variety. It is conjectured that all secant varieties of Segre products of projective spaces are arithmetically Cohen-Macaulay. A summary of the known cases where the conjecture is true is given. An inductive procedure based on the work of Landsberg and Weyman (LW-lifting) is described and used to obtain resolutions of orbits of secant varieties from those of smaller secant varieties. A new computation of the minimal free resolution of the variety of border rank 4 tensors of format…
In projective algebraic geometry, there are classical and fundamental results that describe the structure of geometry and syzygies, and many of them characterize varieties of minimal degree and del Pezzo varieties. In this paper, we consider analogous objects in the category of higher secant varieties. Our main theorems say that there is a matryoshka structure among those basic objects including a generalized $K_{p,1}$ theorem, syzygetic and geometric characterizations of higher secant varieties of minimal degree and del Pezzo higher secant varieties, defined in this paper. For our purpose, we…
We prove that the minimal free resolution of the secant variety of a curve is asymptotically pure. As a corollary, we show that the Betti numbers of converge to a normal distribution.
We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three - with one exception, the secant variety of the $21$-dimensional spinor variety in $\pp{63}$ where we show the ideal is generated in degree four. We also discuss the coordinate rings of secant varieties of compact Hermitian symmetric spaces.
A line bundle L on a smooth curve X is nonspecial if and only if L admits a presentation L=K_X -D +E for some effective divisors D and E>0 on X with gcd (D, E)=0 and h^0 (X, O_X (D))=1. In this work, we define a minimal presentation of L which is minimal with respect to the degree of E among the presentations. If L=K_X -D +E with degE>2 is a minimal, then L is very ample and any q-points of X with q <degE are embedded in general position but the points of E are not. We investigate sufficient conditions on divisors D and E for L=K_X -D +E to be minimal. Through this, for a number n in some rang…
For a vector bundle V over a curve X of rank n and for each integer r in the range 1 \le r \le n-1, the Segre invariant s_r is defined by generalizing the minimal self-intersection number of the sections on a ruled surface. In this paper we generalize Lange and Narasimhan's results on rank 2 bundles which related the invariant s_1 to the secant varieties of the curve inside certain extension spaces. For any n and r, we find a way to get information on the invariant s_r from the secant varieties of certain subvariety of a scroll over X. Using this geometric picture, we obtain a new proof of the…
Let $G$ be a simple algebraic group with $\mathfrak g=\mathsf{Lie} G$ and $\mathcal O_{\sf min}\subset\mathfrak g$ the minimal nilpotent orbit. For a $\mathbb Z_2$-grading $\mathfrak g=\mathfrak g_0\oplus\mathfrak g_1$, let $G_0$ be a connected subgroup of $G$ with $\mathsf{Lie} G_0=\mathfrak g_0$. We study the $G_0$-equivariant projections $φ:\overline{\mathcal O_{\sf min}}\to \mathfrak g_0$ and $ψ:\overline{\mathcal O_{\sf min}}\to\mathfrak g_1$. It is shown that the properties of $\overline{φ(\mathcal O_{\sf min})}$ and $\overline{ψ(\mathcal O_{\sf min})}$ essentially depend on whether the …
Let S be a smooth cubic surface over a field K. It is well-known that new K-rational points may be obtained from old ones by secant and tangent constructions. A Mordell-Weil generating set is a subset B of S(K) of minimal cardinality which generates S(K) via successive secant and tangent constructions. Let r(S,K) be the cardinality of such a Mordell-Weil generating set. Manin posed what is known as the Mordell-Weil problem for cubic surfaces: if K is finitely generated over its prime subfield then r(S,K) is finite. In this paper, we prove a special case of this conjecture. Namely, if S contain…
We give a partial "quasi-stratification" of the secant varieties of the order $d$ Veronese variety $X_{m,d}$ of $\mathbb {P}^m$. It covers the set $σ_t(X_{m,d})^{\dagger}$ of all points lying on the linear span of curvilinear subschemes of $X_{m,d}$, but two "quasi-strata" may overlap. For low border rank two different "quasi-strata" are disjoint and we compute the symmetric rank of their elements. Our tool is the Hilbert schemes of curvilinear subschemes of Veronese varieties. To get a stratification we attach to each $P\in σ_t(X_{m,d})^{\dagger}$ the minimal label of a quasi-stratum containi…
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.
We introduce a relativization of the secant sheaves used by Ein, Green and Lazarsfeld and apply this construction to the study of syzygies of canonical curves. As a first application, we give a simpler proof of Voisin's Theorem for general canonical curves. This completely determines the terms of the minimal free resolution of the coordinate ring of such curves. Secondly, in the case of curves of even genus, we enhance Voisin's Theorem by providing a structure theorem for the last syzygy space, resolving the Geometric Syzygy Conjecture in even genus.
We compute the successive minima of the projective toric variety $X_\cA$ associated to a finite set $ \cA \subset \Z^n$. As a consequence of this computation and of the results of S.-W. Zhang on the distribution of small points, we derive estimates for the height of the subvariety $X_\cA$ and of the $\cA$-resultant. These estimates allow us to obtain an arithmetic analogue of the Bezout-Kushnirenko's theorem concerning the number of solutions of a system of polynomial equations. As an application of this result, we improve the known estimates for the height of the polynomials in the sparse Nul…
Let $X$ be a normal and geometrically integral projective variety over a global field $K$ and let $\overline{D}$ be an adelic Cartier divisor on $X$. We prove a conjecture of Chen, showing that the essential minimum $ζ_{\mathrm{ess}}(\overline{D})$ of $\overline{D}$ equals its asymptotic maximal slope under mild positivity assumptions. As an application, we see that $ζ_{\mathrm{ess}}(\overline{D})$ can be read on the Okounkov body of the underlying divisor $D$ via the Boucksom--Chen concave transform. This gives a new interpretation of Zhang's inequalities on successive minima and a criterion …
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.
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.
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.
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.
In this paper, we shed light on relations between three concepts studied in representations theory, algebraic geometry and quantum information theory. First - spherical actions of reductive groups on projective spaces. Second - secant varieties of homogeneous projective varieties, and the related notions of rank and border rank. Third - quantum entanglement. Our main result concerns the relation between the problem of the state reconstruction from its reduced one-particle density matrices and the minimal number of separable summands in its decomposition. More precisely, we show that sphericity…
Let $\mathbb X\subset\mathbb P(V)$ be a projective variety, which is not contained in a hyperplane. Then every vector $v$ in $V$ can be written as a sum of vectors from the affine cone $X$ over $\mathbb X$. The minimal number of summands in such a sum is called the rank of $v$. The set of vectors of rank $r$ is denoted by $X_r$ and its projective image by $\mathbb X_r$. The r-th secant variety of $X$ is defined $σ_r(\mathbb X):=\bar{\sqcup_{s\le r}\mathbb X_s}$; it is called tame if $σ_r(\mathbb X)=\sqcup_{s\le r} \mathbb X_s$ and wild if the closure contains elements of higher rank. In this p…
Given a line bundle L on a smooth projective curve over the complex numbers, we show that a general extension E of L by the trivial line bundle is very stable: line bundles contained in E have degree much less than half the degree of E. From this result we deduce new inequalities for the successive minima of the euclidean lattice H^1(X,L^{-1}), where L is an hermitian line bundle on the arithmetic surface X.
In this paper we study the distribution of successive minima of global sections of powers of a metrized ample line bundle on a variety over a number field. We develop criteria for there to exist a measure on the real line describing the limiting behavior of this distribution as one considers increasing powers of the bundle. When this measure exists, we develop methods for determining it explicitly. We present applications to the distribution of Petersson norms of cusp forms of increasing weight for SL_2(Z) and to the minimal sup norm of algebraic functions on adelic subsets of curves arising i…
On a given arithmetic surface, inspired by work of Miyaoka, we consider vector bundles which are extensions of a line bundle by another one. We give sufficient conditions for their restriction to the generic fiber to be semi-stable. We then apply the arithmetic analog of Bogomolov inequality in Arakelov theory, and deduce from it a lower bound for some successive minima in the lattice of extension classes between these line bundles.
