Stable modification of relative curves

Stable Modification of Relative Curves

We generalize theorems of Deligne-Mumford and de Jong on semistable modifications of families of proper curves. The main result states that after a generically étale alteration of the base any (not necessarily proper) family of multipointed curves with semi-stable generic fiber admits a minimal semi-stable modification. The latter can also be characterized by the property that its geometric fib...

Relative Compactness and Product Stable Quotient Maps

Stable Maps of Curves

Let h:X → Y be a finite morphism of smooth connected complete curves over Cp. We show h extends to a finite morphism between semi-stable models of X and Y . 2000 Mathematics Subject Classification: 11G20

Relative Bogomolov’s Inequality and the Cone of Positive Divisors on the Moduli Space of Stable Curves

Throughout this paper, we fix an algebraically closed field k. Let f : X → Y be a surjective and projective morphism of quasi-projective varieties over k with dim f = 1. Let E be a vector bundle of rank r on X . Then, we define the discriminant divisor of E with respect to f : X → Y to be disX/Y (E) = f∗ (( 2rc2(E)− (r − 1)c1(E) ) ∩ [X ]) . Here f∗ is the push-forward of cycles, so that disX/Y ...

Spin Curves over Non-stable Curves

Here we consider degenerations of stable spin curves for a fixed smoothing of a non-stable curve: we are able to give enumerative results and a description of limits of stable spin curves. We give a geometrically meaningful definition of spin curves over non-stable curves.