A remark on curves covered by coverings

Abstract

Let f : C ′ → C f:C’ \to C be a covering between two smooth irreducible projective curves. Let p p be a prime number. If C ′ C’ is a covering of degree p p of a curve of genus h h and if g ( C ) ⩾ ( 2 p ) ( h + 3 ) + h + 3 − 2 p g(C) \geqslant (_2^p)(h + 3) + h + 3 - 2p , then C C is a covering of degree p p of a curve of genus at most h h .