FREE DIVISORS IN A PENCIL OF CURVES

Date: 
2015/11/06 Fri 10:30 - 12:00
Room: 
Room 152, Building No.3
Speaker: 
Jean Vallés
Affiliation: 
Universit´e de Pau et des Pays de l’Adour
Abstract: 

According to K. Saito a reduced divisor in an ane or projective space is free if its
module of derivation is a free module. In particular, a plane curve D P2(k) (where k
is supposed to be a eld of characteristic zero in this talk) is free if its associated sheaf
TD of vector elds tangent to D is a free OP2(k)-module. Relatively few free curves are
known. In this talk I prove that the union of all the singular members of a pencil of degree
n plane curves with a smooth base locus (i.e. the base locus consists of n2 distinct points)
is a free divisor and I give its exponents. More generally, I describe the vector bundle
of logarithmic vector elds tangent to any union of curves of the pencil by studying one
particular vector eld \canonically tangent" to the pencil.
This gives already a new and easy method to produce free divisors.