相关论文: New methods for determining speciality of linear s…
Alexander and Hirschowitz determined the Hilbert function of a generic union of fat points in a projective space when the number of fat points is much bigger than the greatest multiplicity of the fat points. Their method is based on a lemma…
The main goal of this paper is to present an algorithm bounding the dimension of a linear system of curves of given degree (or monomial basis) with multiple points in general position. As a result we prove the Hirschowitz--Harbourne…
We consider the vector space of globally differentiable piecewise polynomial functions defined on a three-dimensional polyhedral domain partitioned into tetrahedra. We prove new lower and upper bounds on the dimension of this space by…
We investigate the minimal graded free resolutions of ideals of at most n+1 fat points in general position in P^n. Our main theorem is that these ideals are componentwise linear. This result yields a number of corollaries, including the…
In this paper we deal with linear systems of P^3 through fat points. We consider the behavior of these systems under a cubo-cubic Cremona transformation that allows us to produce a class of special systems which we conjecture to be the only…
We give upper-bounds for the dimension of some linear systems. The theorem improves the differential Horace method introduced by Alexander-Hirschowitz, and was conjectured by Simpson. Possible applications are the calculus of the dimension…
A broadly applicable geometric approach for constructing nef divisors on blow ups of algebraic surfaces at n general points is given; it works for all surfaces in all characteristics for any n. This construction is used to obtain…
The main goal of this paper is to present a new algorithm bounding the regularity and ``alpha'' (the lowest degree of existing hypersurface) of a linear system of hypersurfaces (in $\mathbb P^n$) passing through multiple points in general…
The main result provides an algorithm for determining the minimal free resolution of ideals of fat point subschemes of ${\bf P}^2$ involving up to 8 general points with arbitrary multiplicities; the results hold over algebraically closed…
In this note we consider the behavior of linear systems of P^3 through fat points under a cubo-cubic Cremona transformation. This allows us to produce a class of special systems which we conjecture to be the only ones.
We consider the problem of bounding the dimension of the linear system of curves in ${\bf P}^2$ of degree $d$ with prescribed multiplicities $m_1,...,m_n$ at $n$ general points (\cite{Hir1},\cite{Hir2}). We propose a new method, based on…
We give a closed formula for the dimension of all linear systems in $\mathbb{P}^n$ with assigned multiplicity at arbitrary collections of points lying on a rational normal curve of degree $n$. In particular we give a purely geometric…
By defining a fat point subscheme of $P^2$ to be a 0-dimensional subscheme defined by a sheaf of integrally closed ideals one extends the notion of fat point subschemes to allow infinitely near points. With this notion of fat points, this…
We investigate the expected dimensionality of linear systems with general fat points on certain surfaces using an approach by specialization to elliptic surfaces. For the projectivization of the Atiyah bundle over an elliptic curve with a…
Working over the complex field and formalizing and sharpening approaches introduced by several authors, we give a method for verifying when a divisor on a blow up of P^2 at general points is nef. The method is useful both theoretically and…
We consider linear systems on toric varieties of any dimension, with invariant base points, giving a characterization of special linear systems. We then make a new conjecture for linear systems on rational surfaces.
A sharp bound is obtained for the number of ways to express the monomial $X^n$ as a product of linear factors over $\mathbb{Z}/p^{\alpha}\mathbb{Z}$. The proof relies on an induction-on-scale procedure which is used to estimate the number…
We study divisors in the blow-up of $\mathbb{P}^n$ at points in general position that are non-special with respect to the notion of linear speciality introduced in [5]. We describe the cohomology groups of their strict transforms via the…
We prove Fourier restriction estimates by means of the polynomial partitioning method for compact subsets of any sufficiently smooth hyperbolic hypersurface in threedimensional euclidean space. Our approach exploits in a crucial way the…
Let $\pi: X \rightarrow \mathbb{P}^2$ be the blow-up of $\mathbb{CP}^2$ in $n$ points $x_i$ in very general position, and let $E_i$ be the exceptional divisor over $x_i$. For $0 \leq n \leq 9$ we calculate Okounkov bodies of graded linear…