Alternating Catalan numbers and cover with triple ramification
The Catalan numbers $C_n:=\frac{1}{n+1}\left(\begin{array}{c}2 n \\ n\end{array}\right)$ form one of the most ubiquitous sequence in classical combinatorics. Stanley's book [St] lists 66 different manifestations of these numbers in various counting problems. In the theory of algebraic curves, the Ca...
| Autores: | ; ; ; |
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| Tipo de documento: | Artículo |
| Estado: | Accepted version |
| Fecha de publicación: | 2021 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/190456 |
| Acceso en línea: | https://hdl.handle.net/2445/190456 |
| Nivel de acceso: | Acceso abierto |
| Palabra clave: | Corbes algebraiques Geometria algebraica Teoria de grups Combinatòria (Matemàtica) Algebraic curves Algebraic geometry Group theory Combinations |
| _version_ | 1878738447904014336 |
|---|---|
| author | Farkas, Gavril |
| author2 | Moschetti, Riccardo Naranjo del Val, Juan Carlos Pirola, Gian Pietro |
| author2_role | author author author |
| author_browse | Farkas, Gavril Moschetti, Riccardo Naranjo del Val, Juan Carlos Pirola, Gian Pietro |
| author_facet | Farkas, Gavril Moschetti, Riccardo Naranjo del Val, Juan Carlos Pirola, Gian Pietro |
| author_role | author |
| collection | Dipòsit Digital de la UB |
| description | The Catalan numbers $C_n:=\frac{1}{n+1}\left(\begin{array}{c}2 n \\ n\end{array}\right)$ form one of the most ubiquitous sequence in classical combinatorics. Stanley's book [St] lists 66 different manifestations of these numbers in various counting problems. In the theory of algebraic curves, the Catalan number $C_n$ counts the covers $C \rightarrow \mathbb{P}^1$ of minimal degree $n+1$ from a general curve $C$ of genus $2 n$. Each such cover has simple ramification and its monodromy group equals $S_{n+1}$. By degenerating $C$ to a rational $g$-nodal curve, it was already known to Castelnuovo $[\mathrm{C}]$ that the number of such covers coincides with the degree of the Grassmannian $G(2, n+2)$ in its Plücker embedding, which is well-known to equal $C_n$. |
| eu_rights_str_mv | openAccess |
| format | article |
| id | ES_af2bf4b48bff7344969d7d74e8e6cfce |
| institution | Universidad de Barcelona |
| instname_str | Universidad de Barcelona |
| network_acronym_str | ES |
| network_name_str | España |
| oai_identifier_str | oai:diposit.ub.edu:2445/190456 |
| publishDate | 2021 |
| publishDateSort | 2021 |
| publisher | Centro Edizioni Scuola Normale Superiore di Pisa |
| publisherStr | Centro Edizioni Scuola Normale Superiore di Pisa |
| reponame_str | Dipòsit Digital de la UB |
| spelling | Alternating Catalan numbers and cover with triple ramification Farkas, Gavril Moschetti, Riccardo Naranjo del Val, Juan Carlos Pirola, Gian Pietro Corbes algebraiques Geometria algebraica Teoria de grups Combinatòria (Matemàtica) Algebraic curves Algebraic geometry Group theory Combinations The Catalan numbers $C_n:=\frac{1}{n+1}\left(\begin{array}{c}2 n \\ n\end{array}\right)$ form one of the most ubiquitous sequence in classical combinatorics. Stanley's book [St] lists 66 different manifestations of these numbers in various counting problems. In the theory of algebraic curves, the Catalan number $C_n$ counts the covers $C \rightarrow \mathbb{P}^1$ of minimal degree $n+1$ from a general curve $C$ of genus $2 n$. Each such cover has simple ramification and its monodromy group equals $S_{n+1}$. By degenerating $C$ to a rational $g$-nodal curve, it was already known to Castelnuovo $[\mathrm{C}]$ that the number of such covers coincides with the degree of the Grassmannian $G(2, n+2)$ in its Plücker embedding, which is well-known to equal $C_n$. Centro Edizioni Scuola Normale Superiore di Pisa https://hdl.handle.net/2445/190456 |
| spellingShingle | Alternating Catalan numbers and cover with triple ramification Farkas, Gavril Corbes algebraiques Geometria algebraica Teoria de grups Combinatòria (Matemàtica) Algebraic curves Algebraic geometry Group theory Combinations |
| status_str | acceptedVersion |
| title | Alternating Catalan numbers and cover with triple ramification |
| title_full | Alternating Catalan numbers and cover with triple ramification |
| title_fullStr | Alternating Catalan numbers and cover with triple ramification |
| title_full_unstemmed | Alternating Catalan numbers and cover with triple ramification |
| title_short | Alternating Catalan numbers and cover with triple ramification |
| title_sort | Alternating Catalan numbers and cover with triple ramification |
| topic | Corbes algebraiques Geometria algebraica Teoria de grups Combinatòria (Matemàtica) Algebraic curves Algebraic geometry Group theory Combinations |
| topic_facet | Corbes algebraiques Geometria algebraica Teoria de grups Combinatòria (Matemàtica) Algebraic curves Algebraic geometry Group theory Combinations |
| url | https://hdl.handle.net/2445/190456 |