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The vanishing point theorem is the principal theorem in the science of perspective. It says that the image in a picture plane π of a line L in space, not parallel to the picture, is determined by its intersection with π and its vanishing point. Some authors have used the phrase, "the image of a line includes its vanishing point".
Bregman–Minc inequality (discrete mathematics) Brianchon's theorem ; British flag theorem (Euclidean geometry) Brooks's theorem (graph theory) Brouwer fixed-point theorem ; Browder–Minty theorem (operator theory) Brown's representability theorem (homotopy theory) Bruck–Chowla–Ryser theorem (combinatorics) Brun's theorem (number theory)
In algebraic geometry, a vanishing theorem gives conditions for coherent cohomology groups to vanish. Andreotti–Grauert vanishing theorem; Bogomolov–Sommese vanishing theorem; Grauert–Riemenschneider vanishing theorem; Kawamata–Viehweg vanishing theorem; Kodaira vanishing theorem; Le Potier's vanishing theorem; Mumford vanishing theorem
Beauville–Laszlo theorem; Behrend's trace formula; Belyi's theorem; Bézout's theorem; Birkhoff–Grothendieck theorem; Bogomolov–Sommese vanishing theorem; Borel fixed-point theorem; Borel's theorem
Note that for large m the line bundle K M ⊗ L ⊗m is very ample by the standard Serre's vanishing theorem (and its complex analytic variant). Fujita conjecture provides an explicit bound on m, which is optimal for projective spaces.
A classic example of this is the twisted cubic in : it is a smooth local complete intersection meaning in any chart it can be expressed as the vanishing locus of two polynomials, but globally it is expressed by the vanishing locus of more than two polynomials.
The strange car-chase movie 'Vanishing Point' has had an equally strange afterlife, as detailed in this new book about the film and its star, a 1970 Dodge Challenger R/T 440.
According to the hairy ball theorem, there is a p such that v(p) = 0, so that s(p) = p. This argument breaks down only if there exists a point p for which s(p) is the antipodal point of p, since such a point is the only one that cannot be stereographically projected onto the tangent plane of p.