Number of full exceptional collections modulo spherical twists for elliptic orbifolds

Abstract

This paper calculates the number of full exceptional collections modulo an action of the quotient of a group as the set generated by spherical twists for an abelian category of coherent sheaves over an orbifold projective line with a zero orbifold Euler characteristic. This is done by a recursive formula naturally generalizing for the Dynkin case and an abelian category of coherent sheaves over an orbifold projective line with a positive orbifold Euler characteristic.

Moreover, we have another expression of the degree of the Lyashko-Looijienga map for the universal unfolding of a simple elliptic singularity in the Legendre normal form calculated by Hertling-Roucairol.