AI 中文总结
研究光滑三次曲面在代数闭域上的模域，核心方法是利用模域定义，主要贡献是证得在特征\(0\)时，光滑三次曲面在其模域上有模型。
AI 中文摘要
在代数闭域\(K\)上，簇\(X\)的模域定义为\(K\)中使\(X\cong X^{\sigma}\)的自同构\(\sigma\)的不动域。一个基本问题是簇在何种条件下在其模域上有模型。我们证明，在特征\(0\)时，每个光滑三次曲面在其模域上有模型。
英文摘要
The field of moduli of a variety $X$ over an algebraically closed field $K$ is defined as the fixed field of those automorphisms $σ$ of $K$ for which $X\simeq X^σ$. A fundamental question is under what conditions a variety admits a model over its field of moduli. We give a complete answer for smooth del Pezzo surfaces in characteristic $0$: every smooth del Pezzo surface of degree at least $3$ has a model over its field of moduli, whereas in degrees $1$ and $2$ there exist smooth complex del Pezzo surfaces with field of moduli $\mathbb{R}$ which do not admit a real model.
