Moduli stacks $\overline L_{g,S}$

This paper is a sequel to the paper by A. Losev and Yu. Manin, in which new moduli stacks $\overline L_{g,S}$ of pointed curves were introduced. They classify curves endowed with a family of smooth points divided into two groups, such that the points of the second group are allowed to coincide. The homology of these stacks form components of the extended modular operad whose combinatorial models are further studied in another paper by Losev and Manin. In this paper the basic geometric properties of $\overline L_{g,S}$ are established using the notion of weighted stable pointed curves introduced recently by B. Hassett. The main result is a generalization of Keel's and Kontsevich–Manin's theorems on the structure of $H^*(\overline M_{0,S})$.