Clusters, toric ranks, and 2-ranks of hyperelliptic curves in the wild case
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917482197942272 |
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| author | Fiore, Leonardo Yelton, Jeffrey |
| author_facet | Fiore, Leonardo Yelton, Jeffrey |
| contents | Given a Galois cover $Y \to X$ of smooth projective geometrically connected curves over a complete discrete valuation field $K$ with algebraically closed residue field, we define a semistable model of $Y$ over the ring of integers of a finite extension of $K$ which we call the \emph{relatively stable model} $\Yrst$ of $Y$, and we discuss its properties, focusing on the case when $Y : y^2 = f(x)$ is a hyperelliptic curve viewed as a degree-$2$ cover of the projective line $X := \proj_K^1$. Over residue characteristic different from $2$, it follows from known results that the toric rank (i.e.\ the number of loops in the graph of components) of the special fiber of $\Yrst$ can be computed directly from the knowledge of the even-cardinality clusters of roots of the defining polynomial $f$. We instead consider the ``wild" case of residue characteristic $2$ and demonstrate an analog to this result, showing that each even-cardinality cluster of roots of $f$ gives rise to a loop in the graph of components of the special fiber of $\Yrst$ if and only if the depth of the cluster exceeds some threshold, and we provide a computational description of and bounds for that threshold. As a bonus, our framework also allows us to provide a formula for the $2$-rank of the special fiber of $\Yrst$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19700 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Clusters, toric ranks, and 2-ranks of hyperelliptic curves in the wild case Fiore, Leonardo Yelton, Jeffrey Algebraic Geometry Number Theory 11G20, 11G07, 14H20 Given a Galois cover $Y \to X$ of smooth projective geometrically connected curves over a complete discrete valuation field $K$ with algebraically closed residue field, we define a semistable model of $Y$ over the ring of integers of a finite extension of $K$ which we call the \emph{relatively stable model} $\Yrst$ of $Y$, and we discuss its properties, focusing on the case when $Y : y^2 = f(x)$ is a hyperelliptic curve viewed as a degree-$2$ cover of the projective line $X := \proj_K^1$. Over residue characteristic different from $2$, it follows from known results that the toric rank (i.e.\ the number of loops in the graph of components) of the special fiber of $\Yrst$ can be computed directly from the knowledge of the even-cardinality clusters of roots of the defining polynomial $f$. We instead consider the ``wild" case of residue characteristic $2$ and demonstrate an analog to this result, showing that each even-cardinality cluster of roots of $f$ gives rise to a loop in the graph of components of the special fiber of $\Yrst$ if and only if the depth of the cluster exceeds some threshold, and we provide a computational description of and bounds for that threshold. As a bonus, our framework also allows us to provide a formula for the $2$-rank of the special fiber of $\Yrst$. |
| title | Clusters, toric ranks, and 2-ranks of hyperelliptic curves in the wild case |
| topic | Algebraic Geometry Number Theory 11G20, 11G07, 14H20 |
| url | https://arxiv.org/abs/2403.19700 |