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Learning the semistable reduction theorem for curves I came across some ambiguity in definitions.

Let $k$ be an algebraically closed field, $C$ an algebraic curve over $k$.

(e.g. Liu "Algebraic Geometry and Algebraic Curves", Def. 10.3.1) $C$ is called semistable, if it is reduced and, if it has only ordinary double points as singularities.
(e.g. Harris, Morrison "Moduli of Curves", p.47 or Abbes "Réduction semi-stable des courbes") A connected $C$ as in 1. is called semistable if additionally every irreducible component $E \subset C$ s.t. $E \simeq \mathbb{P}^1_k$ has at least two intersection points with other components of $C$.

My first question:

Which definition describes the semistability in the sense of GIT, i.e. which definition should I take, if I consider a $k$-valued point of the moduli space $M_g$ of algebraic curves of genus $g$ over $k$?

There is a result in Liu "Stable reduction of finite covers of curves", Corollary 2.8 (in the sense of definition 1): Let $R$ be a DVR, $K = Quot(R)$, $X$ geometrically connected smooth projective curve over $K$, $\mathcal X$ a model of $X$ over $R$. Then, after a finite extension of $K$, $\mathcal X$ is dominated by a semistable model $\mathcal X'$ of $X$ over $R$.

My second question:

Is there any hope, that the above proposition is true in the sense of the definition 2?

flag	asked May 7 2012 at 9:25 finite 23●4

2

$M_g$ is the moduli space of smooth curves so neither 1 or 2 gives curves in this space. If you consider the Deligne--Mumford compactification $\overline{M}_g$ then also neither of the two correspond to points in it; instead points correspond to stable curves. If you start with an arbitrary model as in Liu, the Proposition will not hold using definition 2. For example, you can start with a model which satisfies definition 1 but not 2 then there is no way you dominate it by a model satisfying 2. – ulrich May 7 2012 at 9:38

You should probably read D. Mumford, "Stability of projective varieties". – Dan Petersen May 7 2012 at 10:00

Ambiguity in the definition of a semistable curve

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