Fiber product of Riemann surfaces and dessins d'enfants |
Angélica Vega Moreno |
Escuela Colombiana de Ingeniería |
Abstract: |
In the category of sets there is a construction called the fiber product which satisfies certain universality property. Such a construction cannot be realized in any subcategory. For example the fiber product of two Riemann surface can not be again a Riemann surface. For this reason, we will work with the category of singular Riemann surfaces (which contains the Riemann surfaces) for which this kind of construction is possible. |
Let us consider three compact Riemann surfaces \[S_0\] , \[S_1\] and \[S_2\] together with two non- constant holomorphic maps \[\beta_1:S_1\to S_0\] and \[\beta_2:S_2\to S_0\] . The fiber product of the two pairs \[(S_1, \beta_1)\] and \[(S_2, \beta_2)\] is given by \[S_1\times_{(\beta_1, \beta_2)}S_2=\{(z_1, z_2)\in S_1\times S_2:\ \beta_1(z_1)=\beta_2(z_2)\}\] and for it there is a natural function \[\beta:S_1\times_{(\beta_1, \beta_2)}S_2\to S_0\] defined by \[\beta(z_1, z_2)=\beta(z_1)=\beta(z_2)\] . In general, this fiber product might not be irreducible, non-singular or even connected and therefore it might not be a closed Riemann surface. But it is a singular Riemann surface with a finite number of irreducible components \[R_1,\cdots, R_n\] , each one a compact Riemann surface. Actually, in the paper written by R. A. Hidalgo (2011) and titled The fiber product of Riemann surfaces: A Kleinian group point of view at the level of closed Riemann surfaces, Hidalgo showed that any two of the irreducible components of lowest genus of \[S_1\times_{(\beta_1, \beta_2)}S_2\] (if different from one) define isomorphic closed Riemann surfaces (when they have genus one they are isogenous elliptic curves). The fiber product also satisfies the following universal property: |
If \[R\] is a compact Riemann surface and \[p_j:R\to S_j\] for \[j=1,2\] are non-constant holomorphic functions such that \[\beta_1\circ p_1=\beta_2\circ p_2\] then there exists a non-constant holomorphic map \[t:R\to R_k\] for some \[1\leq k\leq n\] such that \[p_j=\pi_j\circ t\] (this determines uniquely \[t(r)=(p_1(r), p_2(r)))\] where \[\pi_j:S_1\times_{(\beta_1, \beta_2)}S_2\to S_j\] is the projection \[\pi_j(z_1, z_2)=z_j\] for \[j=1, 2\] . |
In this talk I will develop the above construction and expose results I have been established recently. A Fuchsian group description of the irreducible components of the fiber product will be given and, as a consequence of this description, if one of the functions \[\beta_j\] is a regular (branched) covering, then all of them are isomorphic Riemann surfaces. In case that the fiber product is connected (for instance, If \[S_0\] has genus zero by results of Fulton and Hansen (1979) in the paper titled: A connectness theorem for projective verieties, with applications to intersections and singularities of mappings), I provide sufficient conditions for the fiber product to be irreducible. Examples will be exhibited as well as an explicit application to dessins d’enfants. |