arXiv 1407.05852014-07-02math.AGmath.OC

实projective varieties的间隙矢量

Gap vectors of real projective varieties

Grigoriy Blekherman, Sadik Iliman, Martina Juhnke-Kubitzke, Mauricio Velasco

摘要：

设$X\subseteq \mathbb{P}^m$是一个全实的、非退化的射影簇，$Γ\subseteq X(\mathbb{R})$是一般点集。设$P$是$X$上的非负二次型的锥，$Σ$是线性型的平方和的锥。我们检查的尺寸的面$P(Γ)$和$Σ(Γ)$组成的形式在$P$和$Σ$，消失在$Γ$。当集合$Γ$的基数在$1,\dots,\rm{codim}(X)$中变化时，$P(Γ)$和$Σ(Γ)$的维数之间的差异定义了$X$的数值不变量，我们称之为X的间隙向量。在这篇文章中，我们开始系统地研究它的基本性质。我们的主要结果是一个公式的间隙矢量的$X$和$X$和它的一般投影的二次亏损的组件。二次亏量是F. L.扎克二次亏和间隙向量之间的关系，使我们能够有效地计算间隙向量的具体品种，以及证明几个一般性质。我们证明了缺口向量弱增加，获得其增长率的上限，并证明这些上限最终实现所有品种。此外，我们还给出了具有最简单间隙向量的簇的一个刻划：证明了对于最小度的簇，间隙向量全同精确地为零，并刻划了间隙向量等于$(0,\dots, 0,1)$的簇.特别是，我们的结果给出了一个新的证明定理的Blekherman，史密斯和Velasco说，有非负二次型，这不是平方和的每一个品种，这是不是最小程度。最后，我们确定的间隙向量的所有Veronese嵌入的$\mathbb{P}^2$，推广工作的前三位作者。

英文摘要：

Let $X\subseteq \mathbb{P}^m$ be a totally real, non-degenerate, projective variety and let $Γ\subseteq X(\mathbb{R})$ be a generic set of points. Let $P$ be the cone of nonnegative quadratic forms on $X$ and let $Σ$ be the cone of sums of squares of linear forms. We examine the dimensions of the faces $P(Γ)$ and $Σ(Γ)$ consisting of forms in $P$ and $Σ$, which vanish on $Γ$. As the cardinality of the set $Γ$ varies in $1,\dots,\rm{codim}(X)$, the difference between the dimensions of $P(Γ)$ and $Σ(Γ)$ defines a numerical invariant of $X$, which we call the gap vector of X. In this article we begin a systematic study of its fundamental properties. Our main result is a formula relating the components of the gap vector of $X$ and the quadratic deficiencies of $X$ and its generic projections. The quadratic deficiency is a fundamental numerical invariant of projective varieties introduced by F. L. Zak. The relationship between quadratic deficiency and gap vectors allows us to effectively compute the gap vectors of concrete varieties as well as to prove several general properties. We prove that gap vectors are weakly increasing, obtain upper bounds for their rate of growth and prove that these upper bounds are eventually achieved for all varieties. Moreover, we give a characterization of the varieties with the simplest gap vectors: We prove that the gap vector vanishes identically precisely for varieties of minimal degree, and characterize the varieties whose gap vector equals $(0,\dots, 0,1)$. In particular, our results give a new proof of the theorem of Blekherman, Smith and Velasco saying that there are nonnegative quadratic forms which are not sums of squares on every variety, which is not of minimal degree. Finally, we determine the gap vector of all Veronese embeddings of $\mathbb{P}^2$, generalizing work of the first three authors.

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