Given a compact Riemann surface ¯ X of genus g and distinct points p and q on ¯ X, we consider the non-compact Riemann surface X := ¯ X \ {q} with basepoint p ∈ X. The extension of mixed Hodge structures associated to the first two steps of π 1 (X, p) is studied. We show that it determines the element (2g q − 2 p − K) in Pic 0 (¯ X), where K represents the canonical divisor of ¯ X as well as the corresponding extension associated to π 1 (¯ X, p). Finally, we deduce a pointed Torelli theorem for punctured Riemann surfaces.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.