Reducibility of the families of 0-dimensional schemes on a variety

Abstract

IfA is a regular local ring of dimensionr>2, over an algebraically closed fieldk, we show that the Hilbert scheme Hilbⁿ A parametrizing ideals of colengthn inA(dim _k A/I=n) has dimension>cn ^2−2/r and is reducible, for alln>c′, wherec andc′ depend only onr. We conclude that ifV is a nonsingular projective variety of dimensionr>2, the Hilbert scheme Hilbⁿ V parametrizing the 0-dimensional subschemes ofV having lengthn, is reducible for alln>c″(r). We may takec″(r) to be

$$102 ifr = 3,25 ifr = 4,35 ifr = 5,and\left( {1 + r} \right)\left( {{{1 + r} \mathord{\left/ {\vphantom {{1 + r} 4}} \right. \kern-\nulldelimiterspace} 4}} \right)ifr > 5.$$

((1))

The result answers in the negative a conjecture of Fogarty [1] but leaves open the question of the conjectured irreducibility of Hilbⁿ A, whereA has dimension 2. Hilbⁿ V is known to be irreducible ifV is a nonsingular surface (Hartshorne forP ₂, and Fogarty [1]). In all cases Hilbⁿ V and Hilbⁿ A are known to be connected (Hartshorne forP _r, and Fogarty [1]). The author is indebted to Hartshorne for suggesting that Hilbⁿ A might be reducible ifr>2.

The proof has 3 steps. We first show that ifV is a variety of dimensionr, then Hilbⁿ V is irreducible only if it has dimensionr n. We then show that ifA is a regular local ring of dimensionr, Hilbⁿ A can be irreducible only if it has dimension (r−1)(n−1). Finally in § 3 we construct a family of graded ideals of colengthn in the local ringA, and having dimensionc′ n^2−2/r. Since for largen this dimension is greater thanr n, and since Hilbⁿ A↪Hilbⁿ V whenA is the local ring of a closed point onV, the proof is complete, except for (1), which follows from § 3, and the monotonicity of (dim Hilbⁿ V−r n) (see (2)).

In § 4, we comment on some related questions.