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| Main Authors: | , |
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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2010.08414 |
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| _version_ | 1866910593499267072 |
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| author | Filipazzi, Stefano Waldron, Joe |
| author_facet | Filipazzi, Stefano Waldron, Joe |
| contents | A conjecture, known as the Shokurov-Kollár connectedness principle, predicts the following. Let $(X,B)$ be a pair, and let $f \colon X \rightarrow S$ be a contraction with $-(K_X + B)$ nef over $S$; then, for any point $s \in S$, the intersection $f^{-1} (s) \cap \mathrm{Nklt}(X,B)$ has at most two connected components, where $\mathrm{Nklt}(X,B)$ denotes the non-klt locus of $(X,B)$. This conjecture has been extensively studied in characteristic zero, and it has been recently settled in that context. In this work, we consider this conjecture in the setup of positive characteristic algebraic geometry. We prove this conjecture holds for threefolds in characteristic $p> 5$, and, under the same assumptions, we characterize the cases in which $\mathrm{Nklt}(X,B)$ fails to be connected. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_08414 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Connectedness principle for $3$-folds in characteristic $p>5$ Filipazzi, Stefano Waldron, Joe Algebraic Geometry A conjecture, known as the Shokurov-Kollár connectedness principle, predicts the following. Let $(X,B)$ be a pair, and let $f \colon X \rightarrow S$ be a contraction with $-(K_X + B)$ nef over $S$; then, for any point $s \in S$, the intersection $f^{-1} (s) \cap \mathrm{Nklt}(X,B)$ has at most two connected components, where $\mathrm{Nklt}(X,B)$ denotes the non-klt locus of $(X,B)$. This conjecture has been extensively studied in characteristic zero, and it has been recently settled in that context. In this work, we consider this conjecture in the setup of positive characteristic algebraic geometry. We prove this conjecture holds for threefolds in characteristic $p> 5$, and, under the same assumptions, we characterize the cases in which $\mathrm{Nklt}(X,B)$ fails to be connected. |
| title | Connectedness principle for $3$-folds in characteristic $p>5$ |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2010.08414 |