Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · alg-geom/9501005

An improved bound for the degree of smooth surfaces in P4 not of general type.

Michele Cook
View original source ↗Read source document ↗Export metadata ↓

Abstract

This is an addendum to the paper of Braun and Fløystad ([BF]) on the bound for the degree of a smooth surface in $\pfour$ not of general type. Using their construction and the regularity of curves in $\pthree$, one may lower the bound a little more. We will prove the following: Theorem. Let $S$ be a smooth surface of degree $d$ in $\pfour$ not of general type. Then either $d \leq 70$ or $S$ lies on a hypersurface in $\pfour$ of degree 5 and $d \leq 80$. Thus $d \leq 80$. We will also indicate how to lower the bound to 76.

Explore related subjects

alg-geommath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michele Cook. 1995-01-16. An improved bound for the degree of smooth surfaces in P4 not of general type.. https://arxiv.org/abs/alg-geom/9501005

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

KEEP EXPLORING

Related papers

arXiv2019

Classification of Varieties with Canonical Curve Section via Gaussian maps on Canonical Curves

Let $C \subset P^{g-1}$ be a smooth canonical curve of genus $g \geq 3$. The purpose of this article is to further develop a method to classify varieties having $C$ as their curve section, using Gaussian map computations. In a previous article a careful analysis of the degeneration to the cone over the hyperplane section was made for _prime_ Fano threefolds, that is Fano threefolds whose Picard group is generated by the hyperplane bundle. In this article we extend this method and classify Fano threefolds of higher index (which still have Picard number one). We are also able to classify Mukai varieties, i.e. varieties of dimension four or more with canonical curve sections.

alg-geom↗
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↗