论文标题
K3表面有两个参与和低PICARD编号
K3 surfaces with two involutions and low Picard number
论文作者
论文摘要
令$ x $为复杂的代数K3度表面$ 2D $，并且Picard Number $ρ$。假设$ x $承认两个通勤的参与：一个全体形态和一个反塑形。在这种情况下，当$ d = 1 $和$ρ\ geq 2 $时，$ρ\ geq 1 $当$ d \ geq 2 $。对于$ d = 1 $，Elsenhans和Jahnel于2008年已经生产了$ \ Mathbb {Q} $定义的第一个示例。 K3表面由Kondō提供，也可以在$ \ Mathbb {Q} $上定义，可用于实现所有$ d \ geq 2 $的最低$ρ= 2 $。在这些注释中，我们在合理数字上构建了K3表面的新示例，以实现$ d = 2,3,4 $的最低$ρ= 2 $。我们还表明，可以使用节点四分之一的表面来实现无限多个$ d $的最低$ρ= 2 $。最后，我们通过证明任何均匀的晶格$ n $排名$ 1 \ leq r \ leq 10 $和签名$（1，r-1）$来加强了莫里森的结果，存在于$ \ mathbb {r} $上定义的K3表面$ y $ y $ y $ y $ y $ y $ y $
Let $X$ be a complex algebraic K3 surface of degree $2d$ and with Picard number $ρ$. Assume that $X$ admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, $ρ\geq 1$ when $d=1$ and $ρ\geq 2$ when $d \geq 2$. For $d=1$, the first example defined over $\mathbb{Q}$ with $ρ=1$ was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kondō, also defined over $\mathbb{Q}$, can be used to realise the minimum $ρ=2$ for all $d\geq 2$. In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum $ρ=2$ for $d=2,3,4$. We also show that a nodal quartic surface can be used to realise the minimum $ρ=2$ for infinitely many different values of $d$. Finally, we strengthen a result of Morrison by showing that for any even lattice $N$ of rank $1\leq r \leq 10$ and signature $(1,r-1)$ there exists a K3 surface $Y$ defined over $\mathbb{R}$ such that $\textrm{Pic} Y_\mathbb{C}=\textrm{Pic} Y \cong N$.
