Normal projective degenerations of rational and ruled surfaces.

An important problem in low dimensional topology is to understand the properties of embedded or immersed surfaces in 4-dimensional manifolds. In this article, we estimate the lower genus bound of closed, connected, smoothly embedded, oriented surfaces in a smooth, closed, connected, oriented 4-manifold with the cohomology algebra of a rational or ruled surface. Our genus bound depends only on the cohomology algebra rather than on the geometric structure of the 4-manifold. It provides evidence for the genus minimizing property of rational and ruled surfaces.

Download Full-text

The new characterization of ruled surfaces corresponding dual Bézier curves

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.7398 ◽

2021 ◽

Author(s):

Muhsin Incesu

Keyword(s):

Ruled Surfaces ◽

Bezier Curves ◽

Bézier Curves

Download Full-text

The surfaces whose prime-sections contain a

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100016583 ◽

1934 ◽

Vol 30 (2) ◽

pp. 170-177 ◽

Cited By ~ 2

Author(s):

J. Bronowski

Keyword(s):

Hyperelliptic Curves ◽

Ruled Surfaces ◽

Minimum Order ◽

Rational Surfaces ◽

Pencil Of Conics

The surfaces whose prime-sections are hyperelliptic curves of genus p have been classified by G. Castelnuovo. If p > 1, they are the surfaces which contain a (rational) pencil of conics, which traces the on the prime-sections. Thus, if we exclude ruled surfaces, they are rational surfaces. The supernormal surfaces are of order 4p + 4 and lie in space [3p + 5]. The minimum directrix curve to the pencil of conics—that is, the curve of minimum order which meets each conic in one point—may be of any order k, where 0 ≤ k ≤ p + 1. The prime-sections of these surfaces are conveniently represented on the normal rational ruled surfaces, either by quadric sections, or by quadric sections residual to a generator, according as k is even or odd.

Download Full-text

FAMILIES OF AFFINE RULED SURFACES: EXISTENCE OF CYLINDERS

Nagoya Mathematical Journal ◽

10.1017/nmj.2016.22 ◽

2016 ◽

Vol 223 (1) ◽

pp. 1-20 ◽

Cited By ~ 1

Author(s):

ADRIEN DUBOULOZ ◽

TAKASHI KISHIMOTO

Keyword(s):

Minimal Model ◽

Finite Extension ◽

Smooth Curve ◽

Ruled Surfaces ◽

Model Program ◽

Minimal Model Program ◽

Generic Fiber ◽

Projective Model ◽

The Family

We show that the generic fiber of a family $f:X\rightarrow S$ of smooth $\mathbb{A}^{1}$-ruled affine surfaces always carries an $\mathbb{A}^{1}$-fibration, possibly after a finite extension of the base $S$. In the particular case where the general fibers of the family are irrational surfaces, we establish that up to shrinking $S$, such a family actually factors through an $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ over a certain $S$-scheme $Y\rightarrow S$ induced by the MRC-fibration of a relative smooth projective model of $X$ over $S$. For affine threefolds $X$ equipped with a fibration $f:X\rightarrow B$ by irrational $\mathbb{A}^{1}$-ruled surfaces over a smooth curve $B$, the induced $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ can also be recovered from a relative minimal model program applied to a smooth projective model of $X$ over $B$.

Download Full-text