Higher-dimensional analogues of stable curves

Abstract

The Minimal Model Program offers natural higher-dimensional analogues of stable n-pointed curves and maps: stable pairs consisting of a projective variety X of dimension ≥ 2 and a divisor B, that should satisfy a few simple conditions, and stable maps f : (X, B) → Y . Although MMP remains conjectural in higher dimensions, in several important situations the moduli spaces of stable pairs, generalizing those of Deligne–Mumford, Knudsen and Kontsevich, can be constructed more directly, and in considerable generality. We review these constructions, with particular attention paid to varieties with group action, and list some open problems.