Mixed volume and an extension of intersection theory of divisors
Mixed volume and an extension of intersection theory of divisors
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混合体积与除数交集理论的推广
DOI：
10.17323/1609-4514-2010-10-2-343-375
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
A. Khovanskii
中科院分区：
文献类型：
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作者：
Kiumars Kaveh;A. Khovanskii
设K（X）是n维不可约簇X上有理函数的所有非零有限维子空间的集合。对于任何n元组L_1，...，L_n在K（X）中，我们定义一个交集索引[L_1，...，L_n]表示方程组f_1 =... = f_n = 0其中每个f_i是来自空间L_i的类属函数。在计算解时，我们忽略了使某个空间L_i中的所有函数都为零的解x以及使某个子空间L_i中至少有一个函数具有极点的解。集合K（X）是关于自然乘法的交换半群。交叉索引[L_1，...，L_n]可以推广到K（X）的Grothendieck群.这给出了除数的交集理论的一个推广。扩展的理论甚至适用于不完全的品种。我们表明，这个相交指数享有所有的主要性质的混合体积的凸体。本文的灵感来自于牛顿多面体理论中的Bernstein-Kushnirenko定理。
Let K(X) be the collection of all non-zero finite dimensional subspaces of rational functions on an n-dimensional irreducible variety X. For any n-tuple L_1,..., L_n in K(X), we define an intersection index [L_1,..., L_n] as the number of solutions in X of a system of equations f_1 = ... = f_n = 0 where each f_i is a generic function from the space L_i. In counting the solutions, we neglect the solutions x at which all the functions in some space L_i vanish as well as the solutions at which at least one function from some subspace L_i has a pole. The collection K(X) is a commutative semigroup with respect to a natural multiplication. The intersection index [L_1,..., L_n] can be extended to the Grothendieck group of K(X). This gives an extension of the intersection theory of divisors. The extended theory is applicable even to non-complete varieties. We show that this intersection index enjoys all the main properties of the mixed volume of convex bodies. Our paper is inspired by the Bernstein-Kushnirenko theorem from the Newton polytope theory.