Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · alg-geom/9509002

Free Resolutions of Fat Point Ideals on $P^2$

Brian Harbourne
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 preprint determines the number of generators in each degree in a minimal homogeneous set of generators for the homogeneous ideal defining any fat point subscheme supported at points of any plane conic, smooth or not. Special cases for points on a cubic are also studied. All work is over an algebraically closed field of arbitrary characteristic.

Explore related subjects

alg-geommath.AG

Keep this discovery

BibTeXRIS

Brian Harbourne. 1998-06-18. Free Resolutions of Fat Point Ideals on $P^2$. https://doi.org/10.1016/s0022-4049(96)00126-0

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2017

Boundedness and $K^2$ for log surfaces

Let $ε, C$ be two positive real numbers, and $\mathcal C \subset \mathbb R$ be a DCC (descending chain condition) set. Let $(X, B = \sum b_j B_j)$ denote a projective surface with an $\mathbb R$-divisor. Then (1) The class $\{X\}$ of surfaces for which there exists a divisor $B$ such that $(X,B)$ is $ε$-log terminal and $-(K_X + B)$ is nef (excluding only those for which at the same time $K_X\equiv 0$, $B=0$, and $X$ has at worst Du Val singularities), is bounded. (2) The set $\{(K_X + B)^2\}$ of squares for the semi log canonical pairs $(X, B)$ with ample $K_X + B$ and $b_j \in \mathcal C$, is a DCC set. (3) The class $\{(X,B)\}$ of pairs such that $(X, B)$ is semi log canonical, $K_X + B$ is ample, $(K_X + B)^2 = C$ and $b_j \in \mathcal C$, is bounded.

alg-geom↗
arXiv2014

Plane Curves with Hyperbolic and C-hyperbolic Complements

The general problem which initiated this work is: What are the quasiprojective varieties which can be uniformized by means of bounded domains in $\cz^n$ ? Such a variety should be, in particular, C--hyperbolic, i.e. it should have a Carathéodory hyperbolic covering. We study here the plane projective curves whose complements are C--hyperbolic. For instance, we show that most of the curves whose duals are nodal or, more generally, immersed curves, belong to this class. We also give explicit examples of irreducible such curves of any even degree d greater or equal 6.

alg-geom↗