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.