761 research outputs found
Finding an ordinary conic and an ordinary hyperplane
Given a finite set of non-collinear points in the plane, there exists a line
that passes through exactly two points. Such a line is called an ordinary line.
An efficient algorithm for computing such a line was proposed by Mukhopadhyay
et al. In this note we extend this result in two directions. We first show how
to use this algorithm to compute an ordinary conic, that is, a conic passing
through exactly five points, assuming that all the points do not lie on the
same conic. Both our proofs of existence and the consequent algorithms are
simpler than previous ones. We next show how to compute an ordinary hyperplane
in three and higher dimensions.Comment: 7 pages, 2 figure
On the number of ordinary conics
We prove a lower bound on the number of ordinary conics determined by a
finite point set in . An ordinary conic for a subset of
is a conic that is determined by five points of , and
contains no other points of . Wiseman and Wilson proved the
Sylvester-Gallai-type statement that if a finite point set is not contained in
a conic, then it determines at least one ordinary conic. We give a simpler
proof of their result and then combine it with a result of Green and Tao to
prove our main result: If is not contained in a conic and has at most
points on a line, then determines ordinary conics.
We also give a construction, based on the group structure of elliptic curves,
that shows that the exponent in our bound is best possible
Voevodsky's conjecture for cubic fourfolds and Gushel-Mukai fourfolds via noncommutative K3 surfaces
In the first part of this paper we will prove the Voevodsky's nilpotence
conjecture for smooth cubic fourfolds and ordinary generic Gushel-Mukai
fourfolds. Then, making use of noncommutative motives, we will prove the
Voevodsky's nilpotence conjecture for generic Gushel-Mukai fourfolds containing
a -plane \G(2,3) and for ordinary Gushel-Mukai fourfolds containing a
quintic del Pezzo surface.Comment: 16 pages, minor mistakes corrected in version
Birationally superrigid cyclic triple spaces
We prove the birational superrigidity and the nonrationality of a cyclic
triple cover of branched over a nodal hypersurface of degree
for . In particular, the obtained result solves the problem of the
birational superrigidity of smooth cyclic triple spaces. We also consider
certain relevant problems.Comment: 43 page
Surfaces with triple points
In this paper we compute upper bounds for the number of ordinary triple
points on a hypersurface in and give a complete classification for degree
six (degree four or less is trivial, and five is elementary). But the real
purpose is to point out the intricate geometry of examples with many triple
points, and how it fits with the general classification of surfaces
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