Surfaces with canonical map of odd degree
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918438537003008 |
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| author | Lopes, Margarida Mendes Pardini, Rita Pignatelli, Roberto |
| author_facet | Lopes, Margarida Mendes Pardini, Rita Pignatelli, Roberto |
| contents | Let $S$ be a smooth complex minimal surface of general type with $p_g:=h^0(K_S)\ge 4$ whose canonical map is generically finite of odd degree $d>1$ onto a surface $Σ$. We assume that the general canonical curve of $S$ is smooth and that $Σ$ is ruled by lines, and we prove:
- $p_g\le d+2$
- $Σ$ is a cone over the rational normal curve of degree $p_g-2$ in ${\mathbb P}^{p_g-1}$
- $p_g=d+2$ can occur only for $d=3,9,11$.
As a byproduct, we refine previous results by Beauville and Xiao by proving that if one drops the assumption that $Σ$ is ruled by lines then $d\le 5$ if $p_g\ge 112$.
The case $d=3$ being completely classified by the first two named authors, we focus on $d=5$, showing that $p_g\le 5$ and that for $p_g=5$ the surface $S$ has a pencil $|C|$ with $C^2=1$ and $K_SC=5$.
These results suggest that the answer to the question whether the surfaces with canonical map of odd degree $d>1$ have bounded invariants could be positive, in sharp contrast with the case of even degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09216 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Surfaces with canonical map of odd degree Lopes, Margarida Mendes Pardini, Rita Pignatelli, Roberto Algebraic Geometry 14J29 primary, 14H45 secondary Let $S$ be a smooth complex minimal surface of general type with $p_g:=h^0(K_S)\ge 4$ whose canonical map is generically finite of odd degree $d>1$ onto a surface $Σ$. We assume that the general canonical curve of $S$ is smooth and that $Σ$ is ruled by lines, and we prove: - $p_g\le d+2$ - $Σ$ is a cone over the rational normal curve of degree $p_g-2$ in ${\mathbb P}^{p_g-1}$ - $p_g=d+2$ can occur only for $d=3,9,11$. As a byproduct, we refine previous results by Beauville and Xiao by proving that if one drops the assumption that $Σ$ is ruled by lines then $d\le 5$ if $p_g\ge 112$. The case $d=3$ being completely classified by the first two named authors, we focus on $d=5$, showing that $p_g\le 5$ and that for $p_g=5$ the surface $S$ has a pencil $|C|$ with $C^2=1$ and $K_SC=5$. These results suggest that the answer to the question whether the surfaces with canonical map of odd degree $d>1$ have bounded invariants could be positive, in sharp contrast with the case of even degree. |
| title | Surfaces with canonical map of odd degree |
| topic | Algebraic Geometry 14J29 primary, 14H45 secondary |
| url | https://arxiv.org/abs/2604.09216 |