Extended Modular Operad

This paper is a sequel to [LoMa] where moduli spaces of painted stable curves were introduced and studied. We define the extended modular operad of genus zero, algebras over this operad, and study the formal differential geometric structures related to these algebras: pencils of flat connections and Frobenius man-ifolds without metric. We focus here on the combinatorial aspects of the picture. Algebraic geometric aspects are treated in [Ma2]. §0. Introduction and summary This paper, together with [Ma2], constitutes a sequel to [LoMa] where some new moduli spaces of pointed curves were introduced and studied. We start with a review of the main results of [LoMa] and then give a summary of this paper. 0.1. Painted stable curves. Let S be a finite set. A painting of S is a partition of S into two disjoint subsets: white W and black B. Let T be a scheme, S a painted set, g ≥ 0. An S–pointed (or labeled) curve of genus g over T consists of the data where (i) π is a flat proper morphism whose geometric fibres C t are reduced and connected curves, with at most ordinary double points as singularities, and g = H 1 (C t , O C t). (ii) x i , i ∈ S, are sections of π not containing singular points of geometric fibres. Such a curve (0.1) is called painted stable, if the normalization of any irreducible component C ′ of a geometric fibre carries ≥ 3 pairwise different special points when C ′ is of genus 0 and ≥ 1 special points when C ′ is of genus 1. Special points are inverse images of singular points and of the structure sections x i. Equivalently, such a normalization has only a finite automorphism group fixing the special points. We get the usual notion of a (family of) stable painted curves, if S consists of only white points.