On the p-rank of singular curves and their smooth models
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909077015101440 |
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| author | Terzi, Sadık |
| author_facet | Terzi, Sadık |
| contents | In this paper, we are concerned with the computation of the $p$-rank and $a$-number of singular curves and their smooth model. We consider a pair $X, X'$ of proper curves over an algebraically closed field $k$ of characteristic $p$, where $X'$ is a singular curve which lies on a smooth projective variety, particularly on smooth projective surfaces $S$ (with $p_g(S) = 0 = q(S)$) and $X$ is the smooth model of $X'$. We determine the $p$-rank of $X$ by using the exact sequence of group schemes relating the Jacobians $J_X$ and $J_{X'}$. As an application, we determine a relation about the fundamental invariants $p$-rank and $a$-number of a family of singular curves and their smooth models. Moreover, we calculate $a$-number and find lower bound for $p$-rank of a family of smooth curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_06901 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the p-rank of singular curves and their smooth models Terzi, Sadık Algebraic Geometry In this paper, we are concerned with the computation of the $p$-rank and $a$-number of singular curves and their smooth model. We consider a pair $X, X'$ of proper curves over an algebraically closed field $k$ of characteristic $p$, where $X'$ is a singular curve which lies on a smooth projective variety, particularly on smooth projective surfaces $S$ (with $p_g(S) = 0 = q(S)$) and $X$ is the smooth model of $X'$. We determine the $p$-rank of $X$ by using the exact sequence of group schemes relating the Jacobians $J_X$ and $J_{X'}$. As an application, we determine a relation about the fundamental invariants $p$-rank and $a$-number of a family of singular curves and their smooth models. Moreover, we calculate $a$-number and find lower bound for $p$-rank of a family of smooth curves. |
| title | On the p-rank of singular curves and their smooth models |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2309.06901 |