Ramified covering maps of singular curves and stability of pulled back bundles

Abstract

Let $f\,:\, X\, \longrightarrow \, Y$ be a generically smooth nonconstant morphism between irreducible projective curves, defined over an algebraically closed field, which is étale on an open subset of Y that contains both the singular locus of Y and the image, in Y, of the singular locus of X. We prove that the following statements are equivalent:

• (1) The homomorphism of étale fundamental groups

  $$\begin{aligned} f_*\,:\, \pi _1^{\textrm{et}}(X) \,\longrightarrow \,\pi _1^{\textrm{et}}(Y) \end{aligned}$$

  induced by f is surjective.

• (2) There is no nontrivial étale covering $\phi \,:\, Y'\, \longrightarrow \, Y$ admitting a morphism $q:\, X\, \longrightarrow \, Y'$ such that $\phi \circ q \,=\, f$.

• (3) The fiber product $X\times _Y X$ is connected.

• (4) $\dim H^0(X,\, f^*f_* \mathcal {O}_X)\,=\, 1$.

• (5) $\mathcal {O}_Y\, \subset \, f_*\mathcal {O}_X$ is the maximal semistable subsheaf.

• (6) The pullback $f^*E$ of every stable sheaf E on Y is also stable.