On curvilinear subschemes of \(\mathbb{P}^ 2\) (Q1321045)

Let \(Z\) be a curvilinear subscheme of \(\mathbb{P}^ 2\), i.e. a 0-dimensional scheme whose embedding dimension at every point of is support is \(\leq 1\). The relevance of such schemes lies in at least two facts: (a) they are the only non-reduced 0-dimensional schemes which lie on nonsingular plane curves i.e. they can be viewed as divisors on some smooth plane curve; (b) the generic non-reduced 0-dimensional subscheme of \(\mathbb{P}^ 2\) is curvilinear. The authors find a bound for the minimum degree of the plane curves on which \(Z\) imposes independent conditions and show that the Hilbert function of \(Z\) is maximal for a ``generic choice'' of \(Z\). They work over an algebraically closed field.