论文标题
通过曲线正方形的椭圆表面家族的明确覆盖
Explicit coverings of families of elliptic surfaces by squares of curves
论文作者
论文摘要
我们表明，对于每个$ n> 0 $，都有一个椭圆表面的家族，这些椭圆表面由属属的正方形$ 2N+1 $覆盖，并且其hodge结构在$ {\ mathbb q}（\ sqrt {-n}）中采取了动作。通过考虑$ n = 3 $的情况，我们表明一个特定的K3表面家族由$ 7 $的广场覆盖。使用它，我们在$ {\ Mathbb p}^4 $的曲线曲线的平方与$ 15 $普通双点的曲线之间构建了对应关系。这给出了这些K3表面的Kuga-Satake-Deligne对应关系的明确证明，其对它们不相关的任何K3表面，然后证明了这些表面正方形的Hodge猜想的证明。我们得出的结论是，这些表面的动机是木村 - 芬特人。我们的分析在曲线的模量空间之间具有其他数据和这些K3表面的模量空间之间的异性等效性，并具有特定的椭圆纤维。
We show that, for each $n>0$, there is a family of elliptic surfaces which are covered by the square of a curve of genus $2n+1$, and whose Hodge structures have an action by ${\mathbb Q}(\sqrt{-n})$. By considering the case $n=3$, we show that one particular family of K3 surfaces are covered by the square of genus $7$. Using this, we construct a correspondence between the square of a curve of genus $7$ and a general K3 surface in ${\mathbb P}^4$ with $15$ ordinary double points up to isogeny. This gives an explicit proof of the Kuga-Satake-Deligne correspondence for these K3 surfaces and any K3 surfaces isogenous to them, and further, a proof of the Hodge conjecture for the squares of these surfaces. We conclude that the motives of these surfaces are Kimura-finite. Our analysis gives a birational equivalence between a moduli space of curves with additional data and the moduli space of these K3 surfaces with a specific elliptic fibration.
