Postulation of general quartuple fat point schemes in P^3

Postulation of general quintuple fat point schemes in P^3

We study the postulation of a general union Y of double, triple, quartuple and quintuple points of P^3. In characteristic 0, we prove that Y has good postulation in degree $d\ge 11$. The proof is based on the combination of the Horace differential lemma with a computer-assisted proof. We also classify the exceptions in degree 9 and 10.

On the postulation of s^d fat points in P^d

In connection with his counter-example to the fourteenth problem of Hilbert, Nagata formulated a conjecture concerning the postulation of r fat points of the same multiplicity in the projective plane and proved it when r is a square. Iarrobino formulated a similar conjecture in any projective space P^d. We prove Iarrobino's conjecture when r is a d-th power. As a corollary, we obtain new counter-examples modeled on those by Nagata.

Nodal curves and postulation of generic fat points on surfaces

Let X be a smooth projective surface. Here we study the postulation of a general union Z of fat points of X, when most of the connected components of Z have multiplicity 2. This problem is related to the existence of "good" families of curves on X, with prescribed singularities, most of them being nodes, and to the cohomology of suitable line bundles on blowing ups of X. More precise statements are obtained in the case X=P^2.

On the postulation of lines and a fat line

In this note we show that the union of $r$ general lines and one fat line in ${\mathbb P}^3$ imposes independent conditions on forms of sufficiently high degree $d$, where the bound on $d$ is independent of the number of lines. This extends former results of Hartshorne and Hirschowitz on unions of general lines, and of Aladpoosh on unions of general lines and one double line.

平面上2-超胖点的假设

Postulation for 2-superfat points in the plane

研究平面上2-超胖点并集的0维概型假设，证明其具有预期的Hilbert函数并解决特殊插值问题。