正则椭圆曲面的无理性次数
Degree of irrationality of properly elliptic surfaces
浏览论文内容
中文总结 AI 辅助
本文研究带截面正则椭圆曲面的无理性次数，给出其上下界，分析其在模空间中的性质，还研究χ(𝒪_S)=0的情况及不带截面的特殊正则椭圆曲面。
中文摘要 AI 辅助
本文研究带有截面的正则椭圆曲面的无理性次数，证明了min{χ(𝒪_S),2gon(C)}≤irr(S)≤2gon(C)，该下界由典范丛公式与Cayley–Bacharach性质得到且是紧的。我们还研究了无理性次数在模空间中的性质：亏格至少为2的曲线上的非常一般的带截面正则椭圆曲面，其无理性次数至少为4；而χ(𝒪_S)=1或2的特殊族的无理性次数为2。最后，我们研究χ(𝒪_S)=0的正则椭圆曲面，证明其一般下界为4，且对任意亏格至少为2的超椭圆曲线C和任意椭圆曲线E，有irr(C×E)=4。本文还包含不带截面的特殊正则椭圆曲面：对Dolgachev曲面，我们排除了非常一般成员的无理性次数为2的情况，并构造了次数为2和3的特殊例子。
英文摘要
In this paper, we study the degree of irrationality of properly elliptic surfaces with a section. We prove $\min\{χ(\mathcal O_S),\,2\operatorname{gon}(C)\} \leq \operatorname{irr}(S) \leq 2\operatorname{gon}(C)$. The lower bound is obtained from the canonical bundle formula and the Cayley--Bacharach property. We show that this bound is sharp. We also study the behavior of the degree of irrationality in moduli. A very general properly elliptic surface with a section over a curve of genus at least two has degree of irrationality at least four, whereas special families with $χ(\mathcal O_S)=1$ or $2$ have degree two. Finally, we investigate properly elliptic surfaces with $χ(\mathcal O_S)=0$, proving a generic lower bound of four and showing that $\operatorname{irr}(C\times E)=4$ for every hyperelliptic curve $C$ of genus at least two and every elliptic curve $E$. Our paper also includes special properly elliptic surfaces without a section. For Dolgachev surfaces, we exclude degree two for a very general member and construct special examples of degrees two and three.
