We study the base locus of the higher fundamental forms of a projective toric variety $X$ at a general point. More precisely we consider the closure $X$ of the image of a map $({\mathbb C}^*)^k\to {\mathbb P}^n$, sending $t$ to the vector of Laurent monomials with exponents $p_0,\dots,p_n\in {\mathbb Z}^k$. We prove that the $m$-th fundamental form of such an $X$ at a general point has non empty base locus if and only if the points $p_i$ lie on a suitable degree-$m$ affine hypersurface. We then restrict to the case in which the points $p_i$ are all the lattice points of a lattice polytope and we give some applications of the above result. In particular we provide a classification for the second fundamental forms on toric surfaces, and we also give some new examples of weighted $3$-dimensional projective spaces whose blowing up at a general point is not Mori dream.

中文翻译：

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复曲面变体高级基本形式的基础位点

我们在一般点研究投影复曲面变量 $X$ 的高级基本形式的基本轨迹。更准确地说，我们考虑地图 $({\mathbb C}^*)^k\to {\mathbb P}^n$ 的图像的闭包 $X$，将 $t$ 发送到具有指数的 Laurent 单项式的向量$p_0,\dots,p_n\in {\mathbb Z}^k$。我们证明，当且仅当点 $p_i$ 位于合适的度 $m$ 仿射超曲面上时，这种 $X$ 在一般点的第 $m$ 个基本形式具有非空基轨迹。然后我们限制点$p_i$ 都是晶格多面体的晶格点的情况，并且我们给出了上述结果的一些应用。特别是我们提供了复曲面上第二基本形式的分类，