论文标题
通过KP方程来表征Jacobians，并通过屈曲和退化的三角体为Kummer品种：代数几何方法
Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: an algebro-geometric approach
论文作者
论文摘要
由于我们在定理4的证明中发现了一个缺陷，因此撤回了本文，声称在复杂的阿贝尔品种上，整个线性束的完整线性系统的基座减少了。错误是在第7页，行$ -14 $中，我们声称多种$ x $上的除数“数学”是线性等同于零的。这是不真实的。例如，这意味着，对于非扭转点$ x $在Abelian Surface $ a $上，让$ e_x $，$ e _ _ { - x} $和$ e_0 $ blow the $ a $ a $ a $ a $ a $ x $，$ -x $，$ -x $，$ 0 $ 0 $的$ 2e_0 $的$ e_0 $ e____________的$ e_0 $是错误的。因此，我们的论文定理4必须被认为未经证实。我们仍然相信它是正确的。我们论文的所有其他论点都是正确的，但不幸的是，它们取决于上述定理4。确切地说，从定理4遵循定理3，断言定义9的方案$σ（x，θ，g）$减少了。本文的其余部分包含shiota定理的代数几何证明，其特征是通过KP方程（第4节）和Krichever的定理来表征Jacobians通过存在于$ \ Mathbb p^g^g-1的Kummer品种嵌入Kummer品种的拐点或退化的三角中。这些证据也是正确的，但它们取决于定理3的较弱版本，即认为，对于方案$σ（x，θ，g）$的编织两个组成部分的断言通常会降低。反过来，这种较弱的定理3版本将按照证据中使用的相同参数，从debarre声称的猜想中，认为在Abelian品种上的完整线性束的完整线性系统的基座通常在编辑二中均大致减少。
This paper is withdrawn since we found a flaw in the proof of Theorem 4, asserting that the base locus of the complete linear system of an ample line bundle on a complex abelian variety is reduced. The error is in page 7, line $ -14$, where we claim that the divisor "mathcal E" on the variety $X$ is linearly equivalent to zero. This is untrue. For instance, it would imply that, for a non-torsion point $x$ on an abelian surface $A$, letting $E_x$, $E_{-x}$, and $E_0$ the exceptional curves in the blow up of $A$ at $x$, $-x$, and $0$, then $2E_0$ is linearly equivalent to $E_x +E_{-x}$, which is easily seen to be false. Therefore Theorem 4 of our paper has to be considered unproven. We still believe that it holds true. All the other arguments of our paper are correct but unfortunately they depend on the above mentioned Theorem 4. To be precise, from Theorem 4 follows Theorem 3, asserting that the scheme $Σ(X,Θ, G)$ of Definition 9 is reduced. The rest of the paper contains algebro-geometric proofs of Shiota's theorem characterizing Jacobians via the KP equation (Section 4), and of Krichever's theorems characterizing Jacobians by the existence of an inflectionary or degenerate trisecant to the Kummer variety embedded in $\mathbb P^{2^g-1}$ (Theorem 18 and Theorem 25). Also these proofs are correct but they depend on a weaker version of Theorem 3, namely on the assertion that the components of codimension two of the scheme $Σ(X,Θ, G)$ are generically reduced. In turn, this weaker version of Theorem 3 would follow, by the same argument used in its proof, from a conjecture by Debarre asserting that the base locus of the complete linear system of an ample line bundle on an abelian variety is generically reduced in codimension two.
