Ramified covering maps of singular curves and stability of pulled back bundles
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| Format: | Preprint |
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2024
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| _version_ | 1866910286291664896 |
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| author | Biswas, Indranil Kumar, Manish Parameswaran, A. J. |
| author_facet | Biswas, Indranil Kumar, Manish Parameswaran, A. J. |
| contents | Let $f : X \rightarrow Y$ 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: \begin{enumerate} \item The homomorphism of étale fundamental groups $$f_* : π_1^{\rm et}(X) \rightarrowπ_1^{\rm et}(Y)$$ induced by $f$ is surjective.
\item There is no nontrivial étale covering $ϕ: Y' \rightarrow Y$ admitting a morphism $q: X \rightarrow Y'$ such that $ϕ\circ q = f$.
\item The fiber product $X\times_Y X$ is connected.
\item $\dim H^0(X, f^*f_* {\mathcal O}_X)= 1$.
\item ${\mathcal O}_Y \subset f_*{\mathcal O}_X$ is the maximal semistable subsheaf.
\item The pullback $f^*E$ of every stable sheaf $E$ on $Y$ is also stable. \end{enumerate} |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_01635 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ramified covering maps of singular curves and stability of pulled back bundles Biswas, Indranil Kumar, Manish Parameswaran, A. J. Algebraic Geometry Let $f : X \rightarrow Y$ 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: \begin{enumerate} \item The homomorphism of étale fundamental groups $$f_* : π_1^{\rm et}(X) \rightarrowπ_1^{\rm et}(Y)$$ induced by $f$ is surjective. \item There is no nontrivial étale covering $ϕ: Y' \rightarrow Y$ admitting a morphism $q: X \rightarrow Y'$ such that $ϕ\circ q = f$. \item The fiber product $X\times_Y X$ is connected. \item $\dim H^0(X, f^*f_* {\mathcal O}_X)= 1$. \item ${\mathcal O}_Y \subset f_*{\mathcal O}_X$ is the maximal semistable subsheaf. \item The pullback $f^*E$ of every stable sheaf $E$ on $Y$ is also stable. \end{enumerate} |
| title | Ramified covering maps of singular curves and stability of pulled back bundles |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2401.01635 |