Saturation theory and very ample Hilbert functions (Q1416697)

Let \(\phi\) be the Hilbert function of a finite subscheme of \( \mathbb P^2\). It is known that the family of subschemes with Hilbert function \(\phi\) is irreducible, and the generic element, \(Z^\phi\), is reduced. Consider the linear system of curves of degree \(d\) passing through \(Z^\phi\), for any given \(d\). We say that \(\phi\) is \(d\)-uniform if this linear system has no fixed component. In this case we define the \(d\)-saturation, \(\overline \phi_d\), of \(\phi\) to be the Hilbert function of the base locus of this linear system (which contains \(Z^\phi\) but may contain more), i.e.\ the Hilbert function of the saturation of the ideal generated by the degree \(d\) component of \(I_{Z^\phi}\). Note that term by term, \(\overline \phi_d\) is greater than or equal to \(\phi\). The main result of this paper shows how to compute \(\overline \phi_d\). As a corollary, the authors characterize those Hilbert functions \(\phi\) and degrees \(d\) for which \(\overline \phi_d = \phi\), i.e.\ the ideal generated by the components of \(I_{Z^\phi}\) of degrees \(\leq d\) is already saturated and defines \(Z^\phi\) itself. The results are similar in flavor to the Cayley-Bacharach theorems, and indeed liaison is an important tool in the proofs. These results are presented very nicely via \(h\)-vectors, which are the first difference of the Hilbert functions, \(\phi\), and are referred to here as ``staircases''. The precise criterion for \(\overline \phi = \phi\) (omitted here) is called the ``one-step property.'' These results can be rephrased as follows. We denote by \({\mathcal L}_{Z^\phi}(d) := \pi^* {\mathcal O}(d)(-E)\) the line bundle on the blow-up of \(\mathbb P^2\) at \(Z^\phi\) whose sections correspond to curves of degree \(d\) passing through \(Z^\phi\). Then \({\mathcal L}_{Z^\phi} (d)\) is globally generated if and only if \(\phi\) is \(d\)-uniform and has the one-step property. The authors present, and answer, the next natural question: Given \(\phi\), what is the least degree \(d\) for which \({\mathcal L}_{Z^\phi}(d)\) is very ample? The answer is somewhat technical.