Genus $2$ pencils on surfaces with $p_g=K^2=1$, envelopes, and conics tangent to plane cubic curves
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| Formato: | Preprint |
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2026
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| _version_ | 1866917310375133184 |
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| author | Catanese, Fabrizio Ruhland, Noah |
| author_facet | Catanese, Fabrizio Ruhland, Noah |
| contents | We consider $(1,1)$-surfaces, namely, minimal compact complex surfaces $S$ with $p_g (S) =K_S^2=1$: for these the bicanonical map is a covering of degree $4$ of the plane $\mathbb{P}^2$.
And we answer a question posed by Meng Chen, whether they can contain a genus 2 pencil (this is the standard reason of failure of birationality of the bicanonical map).
Our main theorem says that those which admit a genus 2 pencil form an irreducible subvariety of codimension $3$ in their moduli space $\frak M_{[1,1]}$; moreover, the general such surface admits exactly $12$ such pencils.
The real fun is to relate this variety to the geometry of pencils of conics in the plane everywhere tangent to a cubic curve and a line. We investigate the corresponding variety $\mathcal{T}$ of triples and provide explicit equations using the classical theory of envelopes: among others, equations given in terms of the Weierstrass normal form of the cubic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03132 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Genus $2$ pencils on surfaces with $p_g=K^2=1$, envelopes, and conics tangent to plane cubic curves Catanese, Fabrizio Ruhland, Noah Algebraic Geometry Complex Variables 14J10, 14J29, 14E05, 14Q10, 32J15, 32Q40 We consider $(1,1)$-surfaces, namely, minimal compact complex surfaces $S$ with $p_g (S) =K_S^2=1$: for these the bicanonical map is a covering of degree $4$ of the plane $\mathbb{P}^2$. And we answer a question posed by Meng Chen, whether they can contain a genus 2 pencil (this is the standard reason of failure of birationality of the bicanonical map). Our main theorem says that those which admit a genus 2 pencil form an irreducible subvariety of codimension $3$ in their moduli space $\frak M_{[1,1]}$; moreover, the general such surface admits exactly $12$ such pencils. The real fun is to relate this variety to the geometry of pencils of conics in the plane everywhere tangent to a cubic curve and a line. We investigate the corresponding variety $\mathcal{T}$ of triples and provide explicit equations using the classical theory of envelopes: among others, equations given in terms of the Weierstrass normal form of the cubic. |
| title | Genus $2$ pencils on surfaces with $p_g=K^2=1$, envelopes, and conics tangent to plane cubic curves |
| topic | Algebraic Geometry Complex Variables 14J10, 14J29, 14E05, 14Q10, 32J15, 32Q40 |
| url | https://arxiv.org/abs/2603.03132 |