Regeneration of the Cremona Map with
Applications to Pencils of Curves and Foliations in 
\[\mathbb{C}P^2\]
 
Xavier Gómez-Mont
CIMAT (Guanajuato, México)
Abstract:
The cremona map 
\[\phi(x_0:x_1:x_2):=(x_0x_1:x_0x_2:x_1x_2)\]
is a birational map of 
\[\mathbb{C}P^2\]
formed with forms of degree
\[2\]
. This implies that if we have an object that has degree
\[d\]
in
\[\mathbb{C}P^2\]
and we pull it back with
\[\phi\]
we will obtain an object of the same type that will have now degree
\[2d\]
, but it is the same on a Zarizki open set. The new object has difficult singularities where the map
\[\phi\]
is not defined or where it is not biregular. We will present some results we have obtained when we regenerate these difficult singularities into several singularities of multiplicity
\[1\]
and give a topological description of the regenerate objects. In particular, we were surprised in seeing what is coming out of the singularities and how it harmonizes with the birational transform of the original object.
The regeneration is carried out with the family of maps 
\[\phi_t(tx_0x_1+tx_2^2:x_0x_2+tx_1:x_1x_2+tx_0^2),\]
which is a 
\[4\]
to 
\[1\]
branched covering of 
\[\mathbb{C}P^2\]
.
We do the regeneration for the pencil of curves 
\[s_0G_0(x_0,x_1,x_2)+s_1G_1(x_0,x_1,x_2)=0/\ (s_0, s_1)\in\mathbb{C}P^2.\]
 We will give a topological description of the regenerate pencil and in particular show that if the original pencial is a Lefschetz pencil, then the fibres of the new pencil are obtained with 4 copies of the original pencil glued together by many small tubes. We show that the monodromy representation of the new pencil is still irreducible, as is the case for the Lefschetz pencil (Deligne).
Similarly, if we begin with a foliation of degree d given by  
\[A_0(x_0,x_1,x_2)dx_0+A_1(x_0,x_1,x_2)dx_1+A_2(x_0,x_1,x_2)dx_2=0\]
 its pullback will be a foliation of degree
\[2d+1\]
which is birational to the original one, and we will see that when it is regenerated, there come out of the difficult singularities
\[3\]
copies of the original foliation in a kaleidoscopic fashion.
This is joint work with Ch. Bonatti (U. Bourgogne), Javier Gallego (U. Complutense) and Manuel Gonz ́alez Villa (CIMAT).