Let $C$ be a projective smooth connected curve over an algebraically closed field $k$. Let $(P,G,p)$ be a triple, where $G$ is a finite $k$-group scheme, $P$ is a $G$-torsor over $C$, $p\in P(k)$ a rational point.

The question is: do we have a triple $(Q,H,q)$ with $Q$ a reduce scheme so that we have a morphism $(h,\rho):(Q,H,q)\to(P,G,p)$ where $\rho: H\to G$ is a group homomorphism and $h: Q\to P$ is a $C$-scheme morphism sending $q\mapsto p$ and interwines the action.

Remark1: The above condition is not always satisfied when $C$ is not smooth. There is a counterexample of a reduced projective connected curve. This counterexample is due to Deligne.

Remark2: For abelian varieties the above condition is always satisfied. In fact, Madhav V.Nori has proved in "The Fundamental Group-Scheme of an Abelian Variety" that for any triple $(P,G,p)$ over an abelian variety $A$ with $p$ lying over $0\in A(k)$, there is an isogeny $n: A\to A$ such that there is a map $(A, A[n], 0)\to (P,G,p)$ in the sense above.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.