AI 中文总结
研究维数≥4的非常一般阿贝尔簇及亏格4的非常一般雅可比簇中满足特定条件的除子性质，利用相关结论证明皮罗拉关于库默纤维化有理截面的猜想。
AI 中文摘要
我们证明，对于维数≥4的非常一般的阿贝尔簇，在\({\rm CH}^2(A)\)中满足\(D^2 = 0\)的除子\(D\in {\rm CH}^1(A)\)是挠的。对于亏格为4的非常一般的雅可比簇也得到了相同结果。然后我们利用第二个结论证明了皮罗拉的一个猜想，即库默纤维化\(K = J / \pm {\rm Id} \to \mathcal{M}_4\)（其中\(J \to \mathcal{M}_4\)是雅可比纤维化）的任何有理截面一定是由两个三次除子的差给出的格里菲斯 - 皮罗拉截面的倍数。
英文摘要
We prove that for a very general abelian variety of dimension $\geq 4$, a divisor $D\in {\rm CH}^1(A)$ that satisfies $D^2=0$ in ${\rm CH}^2(A)$ is of torsion. The same result is also established for a very general Jacobian in genus $4$. We use then the second statement in order to prove a conjecture of Pirola, which states that any rational section of the Kummer fibration $K=J/\pm {\rm Id}\rightarrow \mathcal{M}_4$, where $J\rightarrow \mathcal{M}_4 $ is the Jacobian fibration, must be a multiple of the Griffiths-Pirola section given by the difference of the two trigonal divisors.
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