Geometric Endomorphisms of the Hesse moduli space of elliptic curves
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866914688851247104 |
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| author | Catanese, Fabrizio Sernesi, Edoardo |
| author_facet | Catanese, Fabrizio Sernesi, Edoardo |
| contents | We consider the geometric map $ \mathfrak C$, called Cayleyan, associating to a plane cubic $E$ the adjoint of its dual curve. We show that $ \mathfrak C$ and the classical Hessian map $ \mathfrak H$ generate a free semigroup. We begin the investigation of the geometry and dynamics of these maps, and of the geometrically special elliptic curves: these are the elliptic curves isomorphic to cubics in the Hesse pencil which are fixed by some endomorphism belonging to the semigroup $\mathcal W(\frak H, \frak C)$ generated by $ \frak H, \frak C$. We point out then how the dynamic behaviours of $ \mathfrak H$ and $ \mathfrak C$ differ drastically. Firstly, concerning the number of real periodic points: for $ \mathfrak H$ these are infinitely many, for $ \mathfrak C$ they are just $4$. Secondly, the Julia set of $ \mathfrak H$ is the whole projective line, unlike what happens for all elements of $\mathcal W (\frak H, \frak C)$ which are not iterates of $ \mathfrak H$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_00113 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Geometric Endomorphisms of the Hesse moduli space of elliptic curves Catanese, Fabrizio Sernesi, Edoardo Algebraic Geometry Complex Variables Dynamical Systems 14H52, 14H50, 14H10, 37F10 We consider the geometric map $ \mathfrak C$, called Cayleyan, associating to a plane cubic $E$ the adjoint of its dual curve. We show that $ \mathfrak C$ and the classical Hessian map $ \mathfrak H$ generate a free semigroup. We begin the investigation of the geometry and dynamics of these maps, and of the geometrically special elliptic curves: these are the elliptic curves isomorphic to cubics in the Hesse pencil which are fixed by some endomorphism belonging to the semigroup $\mathcal W(\frak H, \frak C)$ generated by $ \frak H, \frak C$. We point out then how the dynamic behaviours of $ \mathfrak H$ and $ \mathfrak C$ differ drastically. Firstly, concerning the number of real periodic points: for $ \mathfrak H$ these are infinitely many, for $ \mathfrak C$ they are just $4$. Secondly, the Julia set of $ \mathfrak H$ is the whole projective line, unlike what happens for all elements of $\mathcal W (\frak H, \frak C)$ which are not iterates of $ \mathfrak H$. |
| title | Geometric Endomorphisms of the Hesse moduli space of elliptic curves |
| topic | Algebraic Geometry Complex Variables Dynamical Systems 14H52, 14H50, 14H10, 37F10 |
| url | https://arxiv.org/abs/2309.00113 |