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

期刊论文
发表日期：

2024年2月10日

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Indranil Biswas,
Manish Kumar,
A. J. Parameswaran

摘要

Let $f\,:\, X\, \longrightarrow \, Y$ $f : X ⟶ Y$ 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 $\begin{aligned} f_*\,:\, \pi _1^{\textrm{et}}(X) \,\longrightarrow \,\pi _1^{\textrm{et}}(Y) \end{aligned}$ $\begin{matrix} f_{*} : π_{1}^{et} (X) ⟶ π_{1}^{et} (Y) \end{matrix}$ induced by f is surjective.

(2)

There is no nontrivial étale covering $\phi \,:\, Y'\, \longrightarrow \, Y$ $ϕ : Y^{'} ⟶ Y$ admitting a morphism $q:\, X\, \longrightarrow \, Y'$ $q : X ⟶ Y^{'}$ such that $\phi \circ q \,=\, f$ $ϕ \circ q = f$ .

(3)

The fiber product $X\times _Y X$ $X \times_{Y} X$ is connected.

(4)

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

(5)

$\mathcal {O}_Y\, \subset \, f_*\mathcal {O}_X$ $O_{Y} \subset f_{*} O_{X}$ is the maximal semistable subsheaf.

(6)

The pullback $f^*E$ $f^{*} E$ of every stable sheaf E on Y is also stable.