Families of curves and alterations

Abstract

In this article it is shown that any family of curves can be altered into a semi-stable family. This implies that if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> is an excellent scheme of dimension at most 2 and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> is a separated integral scheme of finite type over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> , then <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> can be altered into a regular scheme. This result is stronger then the results of [ Smoothness, semi-stability and alterations to appear in Publ. Math. IHES]. In addition we deal with situations where a finite group acts.