关于二重覆盖与二元二次型的注记
Note on double coverings and binary quadratic forms
摘要：
设E是概型X上的秩二向量丛.以下三种结构被证明是等价的：a）一个本原二次映射q：E --> L，其值在一个可逆模L中。b）一个双重覆盖f：Y --> X，在Y上赋一个可逆模F，加上一个从F到E的直像的同构。c）X上二次射影空间P（E）上的一个有效Cartier因子。从一个观点到另一个观点的段落都是在它们最大的一般性上仔细解决的：我们只需要2在X上可逆。然后我们限制到射影空间P_n，证明了对P_n的任何双覆盖Y，它诱导的Picard群上的同态是同构，如果n？最后将这个结果应用于P_n上秩为2的向量丛上的二次型。
英文摘要：
Let E be a rank two vector bundle on a scheme X. The following three structures are shown to be equivalent : a) A primitive quadratic map q: E --> L, with values in an invertible module L. b) A double covering f: Y --> X endowed with an invertible module F on Y, plus an isomorphism from the direct image of F to E. c) An effective Cartier divisor on the projective space P(E), of degree two over X. The passages from one of these points of view to another are carefully settled in their greatest generality: we only need that 2 is invertible on X. We then restrict to the projective space P_n, and we prove that for any double covering Y of P_n, the homomorphism on Picard groups it induces is an isomorphism if n ? 3; we finally apply this result to quadratic forms on rank two vector bundles on P_n.
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