On the Stratification of Nested Hilbert Schemes

Abstract

We consider nested Hilbert schemes of the type H _{N−1, N} defined by H _{N−1, N}  = {(Z, W) ∈ H ^N−1 × H ^N |Z ⊂ W}. We show that H _{N−1, N} is stratified by irreducible subvarieties of the type H _{Φ, ψ} = {(Z, W) ∈ H _{N−1, N} |Z ∈ H _Φ, W ∈ H _ψ}, where H _Φ is the locally closed subscheme of the Hilbert scheme parameterizing finite schemes with Hilbert function Φ, and we compute the dimension of the strata H _{Φ, ψ}. We also show that H _{N−2, N} is irreducible and compute the dimension of certain strata. The results apply to the classification of globally generated and very ample Hilbert functions.

Key Words:

• Nested Hilbert schemes
• Globally generated line bundles
• Cayley–Bacharach schemes

Mathematics Subject Classification:

• 14M06
• 14N05

Acknowledgments

Notes

^#Communicated by L. Ein.