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Let $Y$ be a smooth proper variety over a field $k$, let $X$ be a smooth variety over $k$, $U\hookrightarrow X$ the complement of a strict normal crossing divisor and $\phi\colon U\to Y$ a map. By Nagata's theorem, there exists a proper binational map $X'\to X$ that is an isomorphism on $U$ such that $\phi$ extends to $X'\to Y$.

In characteristic zero, by Hironaka's theorem we can choose $X'\to X$ to be a sequence of blow-ups in smooth centers. Is anything known in this direction in positive characteristics, without modifying $U$ with alterations? Is the problem easier if the target $Y$ is a curve or a surface?

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    $\begingroup$ If $Y$ is a curve of genus $\geq 1$, $\phi$ extends to a morphism $X\rightarrow Y$... $\endgroup$
    – abx
    Commented May 31, 2023 at 9:46
  • $\begingroup$ Really? I have never seen a result like this, can you explain it to me? $\endgroup$
    – user197402
    Commented May 31, 2023 at 12:09
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    $\begingroup$ This is usually stated with $Y$ an abelian variety. Then this implies the result for any subvariety of an abelian variety, for instance a curve of genus $\geq 1$. $\endgroup$
    – abx
    Commented May 31, 2023 at 13:13
  • $\begingroup$ It also extends to N'eron models of Abelian varieties (it is roughly the defining property of a N'eron model). Sometimes it is called "Weil extension." $\endgroup$
    – Jason Starr
    Commented May 31, 2023 at 13:42
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    $\begingroup$ There is an alternative argument for projective varieties $Y$ for which every morphism $\mathbf P^1\to Y$ is constant (this includes subvarieties of abelian varieties). Indeed, consider the closure $\bar\Gamma$ of the graph $\Gamma_\phi\subseteq X\times Y$, which maps birationally onto $X$ via the first projection $p\colon\bar\Gamma\to X$. When $X$ is smooth, such a map has linearly connected fibres by a result of Murre, which (by assumption on $Y$) are contracted by the second projection $q\colon\bar\Gamma\to Y$. Then $q$ factors through $p$, i.e. $\phi$ is everywhere defined. $\endgroup$
    – R. van Dobben de Bruyn
    Commented May 31, 2023 at 22:16

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