THE FINITE GENERATION OF THE MONOID OF EFFECTIVE DIVISOR CLASSES ON PLATONIC RATIONAL SURFACES

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.

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On moduli of vector bundles on rational surfaces

Archiv der Mathematik ◽

10.1007/bf01271666 ◽

1987 ◽

Vol 49 (3) ◽

pp. 267-272 ◽

Cited By ~ 6

Author(s):

Edoardo Ballico

Keyword(s):

Vector Bundles ◽

Rational Surfaces ◽

Moduli Of Vector Bundles

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Non-branching RCD(0,N) Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups

Canadian Mathematical Bulletin ◽

10.4153/cmb-2015-052-4 ◽

2015 ◽

Vol 58 (4) ◽

pp. 787-798 ◽

Cited By ~ 1

Author(s):

Yu Kitabeppu ◽

Sajjad Lakzian

Keyword(s):

Diameter Growth ◽

Fundamental Groups ◽

Finitely Generated ◽

Finite Generation ◽

Alexandrov Spaces ◽

Geodesic Spaces ◽

Special Case ◽

Linear Diameter

AbstractIn this paper, we generalize the finite generation result of Sormani to non-branching RCD(0, N) geodesic spaces (and in particular, Alexandrov spaces) with full supportmeasures. This is a special case of the Milnor’s Conjecture for complete non-compact RCD(0, N) spaces. One of the key tools we use is the Abresch–Gromoll type excess estimates for non-smooth spaces obtained by Gigli–Mosconi.

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