SINGULAR CURVES WITH LINE BUNDLES L DEFINED OVER ${\mathbb F}_q$ AND WITH H0(L) = H1(L) = 0

Abstract

Here we prove the existence of several pairs (X, L), where X is a geometrically integral projective curve defined over 𝔽q and L is a line bundle on X defined over 𝔽q and with H0(X, L) = H1(X, L) = 0. These examples are obtained using the existence of similar line bundles on the normalization of X, i.e. a case studied by C. Ballet, C. Ritzenthaler and R. Roland.