Degenerations of smooth projective varieties - MathOverflow

Degenerations of smooth projective varieties

2011-02-19T23:34:00Z

Vague question. Is there anything special about degenerations of smooth projective varieties (separating them from arbitrary projective schemes)?

Precise setup. Let $f:X\to Y$ be a projective flat morphism of algebraic schemes (say, over $\mathbb{C}$), where $Y$ is an irreducible variety. Suppose that the generic fiber of $f$ is smooth and connected. Consider special fibers $X_y=f^{-1}(y)$, $y\in Y$. Of course, $X_y$ may be singular, reducible, and/or non-reduced. Zariski's main theorem implies $X_y$ is connected. Can we say anything else?

Specific Questions. Can $X_y$ have embedded components? Can $X_y$ have non-constant regular functions? (Since $X_y$ is connected, such a regular function would have to be nilpotent.)

Reformulation. Consider a Hilbert scheme of closed subschemes in ${\mathbb P}^N$ (with fixed Hilbert polynomial). There is an open subset in it corresponding to smooth connected projective varieties. The question concerns the closure of this set.

Remark. I ran across this in a concrete situation, where the goal is to understand $Rf_*O_X$. But it seems that the question is quite natural, so perhaps the corresponding statements or counterexamples are well known... Any comments would be helpful!

Answer by Allen Knutson for Degenerations of smooth projective varieties

2011-02-20T00:05:15Z

Q1: Sure, you can have embedded components. Project a curve in $P^3$ into a plane. Where the image crosses itself, you get embedded points.

You can improve connected to "set-theoretically equidimensional, and connected in codimension 1".

I think the real moral should be that you shouldn't keep track just of the scheme structure when you're done degenerating a smooth variety, but of e.g. the induced log structure, and consider a moduli space of varieties + that extra structure. I should probably ask a question about that!

Answer by Sándor Kovács for Degenerations of smooth projective varieties

2011-02-20T01:45:30Z

It seems that you are asking about smoothability of singularities. Some singularities are smoothable some are not.

I don't think there is a general criterion that tells you how to decide whether a specific singularity is smoothable or not and it may not be that a singularity that seems bad is necessarily not smoothable while one that looks OK is.

Hypersurface singularities are smoothable. I bet you can see why.

The simplest example I know of an innocent looking non-smoothable singularity is a cone over an abelian variety of dimension at least $2$. The fact that this is not smoothable follows from that it is a Du Bois singularity and hence by a result of Schwede if it were smoothable, the total space would have rational singularities, which is CM, and then all fibers are CM, but this singularity is not. The same argument shows that any projective variety (or variety with isolated singularities) with DB but not CM singularities give examples of what cannot be the limit of smooth projective varieties.

Another set of examples is provided by quotient singularities. They are rigid and hence non-smoothable in dimension at least $3$, but there are two-dimensional quotient singularities that are smoothable, for instance $\big(\mathbb A^2/(x,y)\sim (-x,-y)\big)\simeq (x^2=yz)\subset \mathbb A^3$ is a hypersurface singularity.

If you put more conditions on $f$ that obviously limits further the possible singularities you can get.

EDIT incorporated Karl's comment into the example given.

Answer by Qing Liu for Degenerations of smooth projective varieties

2011-02-20T16:14:08Z

A partial answer: if $X$ is normal and $Y$ is smooth of dimension $1$, then $X_y$ only have constant regular functions ($X\to Y$ is cohomologically flat in relative dimension $0$). This is proved in Raynaud: Spécialisation du foncteur de Picard, Prop. 6.4.2 (use the characteristic 0 hypothesis here, otherwise it is false even when $X$ is also smooth.) It is also proved in the begining of ''Surfaces fibrées en courbes de genre deux'', Lecture Notes in Math. 1137 (1985) by Gang Xiao.

Add: and of course in this situation $X_y$ has no embedded point as $X$ is (S$_2$).