## Graduate Seminar on Advanced Algebra (S4A3) - Stable reduction of curves - Sommersemester 2016

Prof. Dr. Michael Rapoport
Dr. Eugen Hellmann
Kontakt: hellmann (ergänze @math.uni-bonn.de)

Algebraic curves are the basic examples of algebraic geometric objects. The theory of non-singular, as well as singular, curves over an algebraically closed field is well understood.

In this seminar we want to study families of curves and their degenerate fibers. For example the equation

y²=x(x-1)(x-t)

defines a family of curves parametrized by t. In this example it is easy to see that the curves for t=0 and t=1 have a singularity, whereas all other fibers are smooth.

It is more convenient to study families of curves over the spectrum of a discrete valuation ring A with fraction field K. That is, one studies morphisms X → Spec A whose fibers are curves. In this case one says that the special fiber of X is the reduction of the generic fiber XK of X. If the special fiber of X → Spec A is non-singular, one says that X is a smooth model of XK.

It turns out that in general a non-singular curve XK over Spec K does not admit a smooth model over Spec A. However, there is a notion of a stable model, where the special fiber only has mild singularities.

In this seminar we want to prove the Theorem of Deligne and Mumford that every curve over K admits a stable model after extending scalars to a finite extension. The seminar follows the approach of Artin and Winters.

## Prerequisites

We assume familiarity with the basic concepts of Algebraic Geometry, roughly in the amount of chapters II of Hartshorne's book. The later talks also make use of the contents of chapter III.

## Time and Place

Tuesday 16-18h, MZ Room 0.006

## Program

The detailed program can be found here.

## Organizational meeting

Tuesday 09.02.2015 at 10h (c.t.), MZ Room 0.006.
If you can not come to the organizational meeting please e-mail in advance.

## References

• M. Artin, G. Winters: Degenerate fibers and stable reduction of curves, Topology, vol. 10, pp. 373-383, 1971.
• P. Deligne: Intersection sur les surfaces réguliàres, SGA 7, Expose X.
• R. Hartshorne: Algebraic Geometry, GTM, vol. 52, Springer.
• S. Lichtenbaum: Curves over discrete valuation rings, Am. J. Math., vol. 90, pp. 380-409, 1968.
• Q. Liu: Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Math.