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Title: | Alternating Catalan numbers and cover with triple ramification |
Author: | Farkas, Gavril Moschetti, Riccardo Naranjo del Val, Juan Carlos Pirola, Gian Pietro |
Keywords: | Corbes algebraiques Geometria algebraica Teoria de grups Combinatòria (Matemàtica) Algebraic curves Algebraic geometry Group theory Combinations |
Issue Date: | 24-Jun-2021 |
Publisher: | Centro Edizioni Scuola Normale Superiore di Pisa |
Abstract: | 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 $C_n$. |
Note: | Versió postprint del document publicat a: https://doi.org/10.2422/2036-2145.201909_009 |
It is part of: | Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, 2021, vol. XXII, num. 2, p. 665-690 |
URI: | http://hdl.handle.net/2445/190456 |
Related resource: | https://doi.org/10.2422/2036-2145.201909_009 |
ISSN: | 0391-173X |
Appears in Collections: | Articles publicats en revistes (Matemàtiques i Informàtica) |
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