3,192 research outputs found
Changing Views on Curves and Surfaces
Visual events in computer vision are studied from the perspective of
algebraic geometry. Given a sufficiently general curve or surface in 3-space,
we consider the image or contour curve that arises by projecting from a
viewpoint. Qualitative changes in that curve occur when the viewpoint crosses
the visual event surface. We examine the components of this ruled surface, and
observe that these coincide with the iterated singular loci of the coisotropic
hypersurfaces associated with the original curve or surface. We derive
formulas, due to Salmon and Petitjean, for the degrees of these surfaces, and
show how to compute exact representations for all visual event surfaces using
algebraic methods.Comment: 31 page
Invariant Homology on Standard Model Manifolds
Torus-fibered Calabi-Yau threefolds Z, with base dP_9 and fundamental group
pi_1(Z)=Z_2 X Z_2, are reviewed. It is shown that Z=X/(Z_2 X Z_2), where X=B
X_{P_1} B' are elliptically fibered Calabi-Yau threefolds that admit a freely
acting Z_2 X Z_2 automorphism group. B and B' are rational elliptic surfaces,
each with a Z_2 X Z_2 group of automorphisms. It is shown that the Z_2 X Z_2
invariant classes of curves of each surface have four generators which produce,
via the fiber product, seven Z_2 X Z_2 invariant generators in H_4(X,Z). All
invariant homology classes are computed explicitly. These descend to produce a
rank seven homology group H_4(Z,Z) on Z. The existence of these homology
classes on Z is essential to the construction of anomaly free, three family
standard-like models with suppressed nucleon decay in both weakly and strongly
coupled heterotic superstring theory.Comment: 57 pages, 13 figure
Automorphisms of order three on numerical Godeaux surfaces
We prove that a numerical Godeaux surface cannot have an automorphism of
order three
Divisors class groups of singular surfaces
We compute divisors class groups of singular surfaces. Most notably we
produce an exact sequence that relates the Cartier divisors and almost Cartier
divisors of a surface to the those of its normalization. This generalizes
Hartshorne's theorem for the cubic ruled surface in P^3. We apply these results
to limit the possible curves that can be set-theoretic complete intersection in
P^3 in characteristic zero
- …