Stability of pencils of plane sextics and Halphen pencils of index two

Abstract

We study the stability of pencils of plane sextics in the sense of geometric invariant theory. In particular, we obtain a complete and geometric description of the stability of Halphen pencils of index two.

The moduli space of rational elliptic surfaces of index two

“…The addition of the additional data of the chosen bisection -the marking -seems to rigidify things so that all the RES's of index two are semi-stable. This is not the case for the unmarked analysis ( [10]) nor was the case for the RES's of index one (the Weierstrass fibrations considered in [9]).…”

“…When the choice of a marked bisection is not part of the classification problem, then a possible approach to constructing the corresponding moduli space has been considered in [10]. The addition of the additional data of the chosen bisection -the marking -seems to rigidify things so that all the RES's of index two are semi-stable.…”