Reducibility of the families of 0-dimensional schemes on a variety
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The result answers in the negative a conjecture of Fogarty  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 ). In all cases HilbnV and HilbnA are known to be connected (Hartshorne forPr, and Fogarty ). 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.
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