Inventiones mathematicae

, Volume 15, Issue 1, pp 72–77

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

• A. Iarrobino
Article

## Abstract

IfA is a regular local ring of dimensionr>2, over an algebraically closed fieldk, we show that the Hilbert scheme HilbnA parametrizing ideals of colengthn inA(dimkA/I=n) has dimension>cn2−2/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 HilbnV 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 HilbnA, whereA has dimension 2. HilbnV is known to be irreducible ifV is a nonsingular surface (Hartshorne forP2, and Fogarty [1]). In all cases HilbnV and HilbnA are known to be connected (Hartshorne forPr, and Fogarty [1]). The author is indebted to Hartshorne for suggesting that HilbnA might be reducible ifr>2.

The proof has 3 steps. We first show that ifV is a variety of dimensionr, then HilbnV is irreducible only if it has dimensionr n. We then show that ifA is a regular local ring of dimensionr, HilbnA 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′ n2−2/r. Since for largen this dimension is greater thanr n, and since HilbnA↪HilbnV 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 HilbnV−r n) (see (2)).

In § 4, we comment on some related questions.

## Preview

### References

1. 1.
Fogarty, J.: Algebraic families on an algebraic surface. AJM90, 511–521 (1968).Google Scholar
2. 2.
Hartshorne, R.: Connectedness of the Hilbert scheme. Publ. Math. de I.H.E.S.29, 261–304 (1966).Google Scholar
3. 3.
Iarrobino, A.: Families of ideals in the ring of power series in two variables, Ph.D. Thesis, MIT, Cambridge, MA (1970) (to appear).Google Scholar
4. 4.
Iarrobino, A.: An algebraic bundle overP 1 that is not a vector bundle (to appear).Google Scholar
5. 5.
Iarrobino, A.: The type of an ideal in a local ring (to appear).Google Scholar
6. 6.
Mumford, D.: Geometric invariant theory. Ergebnisse der Mathematik, N. F. Bd. 34. Berlin-Heidelberg-New York: Springer 1965.Google Scholar
7. 7.
Zariski, O., Samuel, P.: Commutative algebra, vol. II. Princeton: Von Nostrand 1960.Google Scholar