Symmetric products of an algebraic curve
Symmetric products of an algebraic curve
复制标题
代数曲线的对称积
DOI：
10.1016/0040-9383(62)90019-8
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发表时间：
1962
期刊：
影响因子：
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通讯作者：
I. G. MacDonald
中科院分区：
文献类型：
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作者：
I. G. MacDonald
设X是复数域C上的一条非奇异不可约的完全代数曲线，设X (n)表示X的第n个对称积。本文第一部分给出了‘ @ ’的整上同环的显式确定，以及X (n)的许多拓扑不变量和代数不变量的计算。一旦在X上固定了一个基点，X (n)就有一个自然映射到X的雅可比矩阵J中，众所周知，如果n> 2g- 2x (n)以这种方式成为J上的一个射影纤维束。在J上有一个标准定义的秩为n-g+ 1的向量束Y，它的相关射影束是X (n)，我们确定了I '和X (n)的陈氏类。接下来，我们将H*（X (n)， Z）的知识应用于关于X的枚举几何的各种问题；特别地，我们得到了早先一篇论文的结果的自然证明，这些结果是用经典方法费力地得到的。最后，我们计算了X (n)的zeta函数（X现在定义在一个有限域上），并在这种情况下验证了Weil的猜想。
LET X be a nonsingular irreducible complete algebraic curve over the field C of complex numbers, and let X (n) denote the nth symmetric product of X. The first part of this paper is devoted to an explicit determination of the integral cohomology ring of’@), and to the calculation of many of the topological and algebraic invariants of X (n). Once a base point has been fixed on X there is a natural mapping of X (n) into the Jacobian J of X, and it is well known that if n> 2g-2 X (n) becomes in this way a proj ective fibre bundle over J. There is a canonically defined vector bundle Y of rank n-g+ 1 on J whose associated projective bundle is X (n), and we determine the Chern classes of I’, and of X (n).Next we apply our knowledge of H*(X (n), Z) to various questions of enumerative geometry on X; in particular we obtain natural proofs of the results of an earlier paper [5] which were there obtained laboriously by classical methods. Finally we calculate the zeta function of X (n)(X being now defined over a finite field) and verify Weil’s conjectures in this case.