对称幂上的有理点与范畴表示
Rational points on symmetric powers and categorical representability
摘要：
在本文中，我们观察到，对于几何整投射簇$X$，允许一个完整的弱例外集组成的纯向量丛，存在一个$k$ -有理点蕴涵$\mathrm{rdim}(X)=0$。研究了$\mathbb{R}$上的Brauer-Severi对称幂$S^n(X)$和对合簇，证明了等变导范畴$D^b_{S_n}(X^n)$存在一个完全弱例外集.因此，我们发现$\mathrm{rdim}(X)=0$当且仅当$\mathrm{rdim}(D^b_{S_n}(X^n))=0$对于$1\leq n\leq 3$。如果$X$是Brauer-Severi，则在$X$或$S^3(X)$上存在$\mathbb{R}$ -有理点等价于$\mathrm{rdim}(D^b_{S_3}(X^3))=0$。
英文摘要：
In this paper we observe that for geometrically integral projective varieties $X$, admitting a full weak exceptional collection consisting of pure vector bundles, the existence of a $k$-rational point implies $\mathrm{rdim}(X)=0$. We also study the symmetric power $S^n(X)$ of Brauer--Severi and involution varieties over $\mathbb{R}$ and prove that the equivariant derived category $D^b_{S_n}(X^n)$ admits a full weak exceptional collection. As a consequence, we find $\mathrm{rdim}(X)=0$ if and only if $\mathrm{rdim}(D^b_{S_n}(X^n))=0$ for $1\leq n\leq 3$. If $X$ is Brauer--Severi, the existence of a $\mathbb{R}$-rational point on $X$ or $S^3(X)$ is equivalent to $\mathrm{rdim}(D^b_{S_3}(X^3))=0$.
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