Related papers: Postulation for 2-superfat points in the plane
Two approaches for determining Hilbert functions of fat point subschemes of $\mathbb P^2$ are demonstrated. A complete determination of the Hilbert functions which occur for 9 double points is given using the first approach, extending…
It remains an open problem to classify the Hilbert functions of double points in $\mathbb{P}^2$. Given a valid Hilbert function $H$ of a zero-dimensional scheme in $\mathbb{P}^2$, we show how to construct a set of fat points $Z \subseteq…
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…
We consider 0-dimensional schemes supported at a single point in n-space that are m-symmetric, i.e. that intersect any smooth curve passing through the point with length m, and the ones among them that are maximal with respect to inclusion…
We study the postulation of a general union $Y$ of double, triple, and quartuple points of $\mathbb{P}^3$. We prove that $Y$ has the expected postulation in degree $d\ge 41$, using the Horace differential lemma. We also discuss the cases of…
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…
In this paper we address the postulation problem of zero-dimensional schemes on a surface of length at most 4. We prove some general results and then we focus on the case of P2, P1xP1 and Hirzebruch surfarces. In particular, we prove that…
We study the postulation of a general union $X\subset \mathbb {P}^3$ of one m-point $mP$ and $t$ disjoint lines. We prove that it has the expected Hilbert function, proving a conjecture by E. Carlini, M. V. Catalisano and A. V. Geramita.
The Hilbert scheme of $n$ points in the affine plane contains the open subscheme parametrizing $n$ distinct points in the affine plane, and the closed subscheme parametrizing ideals of codimension $n$ supported at the origin of the affine…
A recent paper by the first and third authors together with Sabourin raised the question of what the possible Hilbert functions are for fat point subschemes of the form $2p_1+...+2p_r$, for all possible choices of $r$ distinct points in the…
In this paper we find an algorithm which computes the Hilbert function of schemes $Z$ of "fat points" in $\PP3$ whose support lies on a rational normal cubic curve $C$. The algorithm shows that the maximality of the Hilbert function in…
In order to determine the Hilbert function of the ideal of a fat point subscheme of projective space, we show that it is enough to determine, both for the subscheme itself and the subschemes obtained from it by successively adjoining to it…
We relate Hilbert schemes of points and Fulton-MacPherson compactifications by an interpolating stability condition. We then derive wall-crossings formulas and some applications for the enumerative geometry of Hilbert schemes.
We study the bi-graded Hilbert function of ideals of general fat points with same multiplicity in $\mathbb{P}^1\times\mathbb{P}^1$. Our first tool is the multiprojective-affine-projective method introduced by the second author in previous…
The Hilbert scheme $S^{[n]}$ of points on an algebraic surface $S$ is a simple example of a moduli space and also a nice (crepant) resolution of singularities of the symmetric power $S^{(n)}$. For many phenomena expected for moduli spaces…
We study Hilbert functions of certain non-reduced schemes A supported at finite sets of points in projective space, in particular, fat point schemes. We give combinatorially defined upper and lower bounds for the Hilbert function of A using…
The purpose of the present note is to provide a new proof ot the well-known result due to Hartshorne and Hirschowitz to the effect that general lines in projective spaces have good postulation. Our approach uses specialization to a…
We consider the open problem of determining the graded Betti numbers for fat point subschemes supported at general points of the projective plane. We relate this problem to the open geometric problem of determining the splitting type of the…
We study the connection between the generation of a fat point scheme supported at general points in the plane and the behaviour of the cotangent bundle with respect to some rational curves particularly relevant for the scheme. We put…
This is an appendix to the recent paper of Favacchio and Guardo. In these notes we describe explicitly a minimal bigraded free resolution and the bigraded Hilbert function of a set of 3 fat points whose support is an almost complete…