The diagonal set of the self-product of an algebraic curve

Abstract

Let C be a smooth projective curve of genus g defined over an algebraically closed field of characteristic $p\ne 2$ and let Δ be the diagonal of $C\times C$. We observe the complement $X:=(C\times C)\setminus \Delta$. If $g=0$, X is an affine hypersurface $xy=z^2-1$ in $A^3$ which is the simplest example of Danielewski surfaces. If $g=1$ then Δ is a fiber of an elliptic fibration over C so that $\left({\Delta }^2\right)=0$ and $\bar{\kappa }\left(X\right)=1$, and if $g>1$, $\left({\Delta }^2\right)=2-2g$ and Δ is contractible. In the case C has genus $g>1$, X is embedded bijectively into the Jacobian variety if C is non-hyperelliptic, though X is generically a double covering of a surface in the Jacobian variety if C is hyperelliptic. Some observations will be made in the case k has characteristic 2.