Computing the Geometric Intersection Number of Curves

Let X be a smooth projective variety of dimension n over the field of complex numbers. We denote by Kx the canonical bundle of X. By Mori's theory, if Kx is not numerically effective (i.e. if there exists a curve on X which has negative intersection number with Kx), then there exists an extremal ray ℝ+[C] on X and an elementary contraction fR: X → Y associated with ℝ+[C].fR is called a small contraction if it is bi-rational and an isomorphism in co-dimension one.

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SOME NUMERICAL INVARIANTS OF FLAT VIRTUAL LINKS ARE ALWAYS REALIZED BY REDUCED DIAGRAMS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216506004476 ◽

2006 ◽

Vol 15 (03) ◽

pp. 289-297 ◽

Cited By ~ 4

Author(s):

TERUHISA KADOKAMI

Keyword(s):

Intersection Number ◽

Finite Sequence ◽

Closed Surface ◽

Crossing Number ◽

The Self ◽

Virtual Link ◽

Virtual Knot ◽

Virtual Links ◽

Geodesic Loop ◽

Numerical Invariants

Any flat virtual link has a reduced diagram which satisfies a certain minimality, and reduced diagrams are related one another by a finite sequence of a certain Reidemeister move. The move preserves some numerical invariants of diagrams. So we can define numerical invariants for flat virtual links. One of them, the crossing number of a flat virtual knot K, coinsides with the self-intersection number of K as an essential geodesic loop on a hyperbolic closed surface. We also show an equation among these numerical invariants, basic properties by using the equation, and determine non-split flat virtual links with the crossing number up to three.

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Towards the weighted Bounded Negativity Conjecture for blow-ups of algebraic surfaces

manuscripta mathematica ◽

10.1007/s00229-019-01157-2 ◽

2019 ◽

Vol 163 (3-4) ◽

pp. 361-373

Author(s):

Roberto Laface ◽

Piotr Pokora

Keyword(s):

Line Bundle ◽

Characteristic Zero ◽

Algebraic Surface ◽

Intersection Number ◽

Algebraic Surfaces ◽

The Self ◽

Projective Surface ◽

Intersection Numbers ◽

Weighted Version ◽

Smooth Projective Surface

AbstractIn the present paper we focus on a weighted version of the Bounded Negativity Conjecture, which predicts that for every smooth projective surface in characteristic zero the self-intersection numbers of reduced and irreducible curves are bounded from below by a function depending on the intesection of curve with an arbitrary big and nef line bundle that is positive on the curve. We gather evidence for this conjecture by showing various bounds on the self-intersection number of curves in an algebraic surface. We focus our attention on blow-ups of algebraic surfaces, which have so far been neglected.

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