Degeneration (algebraic geometry)

In the study of moduli of curves, the important point is to understand the boundaries of the moduli, which amounts to understand degenerations of curves. == Stability of invariants ==

Ruled-ness specializes. Precisely, Matsusaka'a theorem says :Let X be a normal irreducible projective scheme over a discrete valuation ring. If the generic fiber is ruled, then each irreducible component of the special fiber is also ruled. == Infinitesimal deformations ==

Let D = k[ε] be the ring of dual numbers over a field k and Y a scheme of finite type over k. Given a closed subscheme X of Y, by definition, an embedded first-order infinitesimal deformation of X is a closed subscheme X of Y ×Spec(k) Spec(D) such that the projection is flat and has X as the special fiber. If Y = Spec A and are affine, then an embedded infinitesimal deformation amounts to an ideal of A[ε] such that is flat over D and the image of in A = A[ε]/ε is . In general, given a pointed scheme (S, 0) and a scheme X, a morphism of schemes : X → S is called the deformation of a scheme X if it is flat and the fiber of it over the distinguished point 0 of S is X. Thus, the above notion is a special case when S = Spec D and there is some choice of embedding. == See also ==