Modular operads of embedded curves (Q520877)

In the famous article [Compos. Math. 110, No. 1, 65--126 (1998; Zbl 0894.18005)] \textit{E. Getzler} and \textit{M. Kapranov} studied operadic type structures related to the moduli space of algebraic curve. Intuitively, algebraic curves with marked points can be glued along the marked points generating operations on the moduli spaces. However, when considering arbitrary genuses of curves, the classical operadic picture, in which operations are labeled by trees, is replaced by operation labeled instead by graphs. They call this operadic structure modular. Moreover, moduli of curves with marked points do have typically (for instance when considering genus \(0\) curves) an extra cyclic symmetry obtained by permuting the punctures. In the same paper, they show how to construct a Feynman transform on the category of dg-modular operads and how to compute its Euler characteristic in terms of the Wick's theorem, hence highlighting the relation of this operad with mathematical physics. In this paper the authors present a generalization of these results for curves with marked points \(k-\log\) canonically embedded, meaning admitting a projective embedding by a complete linear system. The study of log canonical models for curves has been central in the study of moduli spaces of curves and for its relationships to the Minimal Model program. There are three results presented: first, they show that for \(k\geq 5\) the moduli spaces of \(k-\log\) canonically embedded curves assemble together in a modular operad in Deligne-Mumford stacks. Second, they show that for \(k\geq 1\) the moduli spaces of \(k-\log\) canonically embedded curves of genus \(0\) assemble together in a cyclic operad in schemes. Third, they show that for \(k\geq 2\) the moduli spaces of \(k-\log\) canonically embedded curves assemble together in a stable cyclic operad in Deligne-Mumford stacks. In order to prove these results, they construct morphisms on these moduli spaces corresponding to the gluing of two embedded curves and to the gluing of two points together on the same embedded curve. The proofs of these statements appear correct. Would be interesting, as a follow up work, to understand weather the construction of Getzler and Kapranov of the Feynman transform could be generalized to this setting.