2015/09/24 by Alexander Polishchuk, Polishchuk, Alexander · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG
paper · pdf · doi:10.48550/arxiv.1509.07241
63 pages, first draft: comments are welcome
arxiv created 2015/09/24 · arxiv updated 2015/09/25
We study moduli spaces of (possibly non-nodal) curves (C,p1,…,pn) of arithmetic genus g with n smooth marked points, equipped with nonzero tangent vectors, such that \mathcal OC(p1+…+pn) is ample and H1(\mathcal OC(a1p1+…+anpn))=0 for given weights \bf a=(a1,…,an) such that ai≥ 0 and ∑ ai=g. We show that each such moduli space \widetilde\mathcal Unsg,n(\bf a) is an affine scheme of finite type, and the Krichever map identifies it with the quotient of an explicit locally closed subscheme of the Sato Grassmannian by the free action of the group of changes of formal parameters. We study the GIT quotients of \widetilde\mathcal Unsg,n(\bf a) by the natural torus action and show that some of the corresponding stack quotients give modular compactifications of \mathcal Mg,n with projective coarse moduli spaces. More generally, using similar techniques, we construct moduli spaces of curves with chains of divisors supported at marked points, with prescribed number of sections, which in the case n=1 corresponds to specifying the Weierstrass gap sequence at the marked point.