140 research outputs found
Ambient Surfaces and T-Fillings of T-Curves
T-curves are piecewise linear curves which have been used with success since
the beginning of the 1990's to construct new real algebraic curves with
prescribed topology mainly on the real projective plane. In fact T-curves can
be used on any real projective toric surface. We generalize here the
construction of the latter by departing from non-convex polygons and we get a
class of surfaces, the ambient surfaces, that we characterize topologically and
on which T-curves are also well defined. Associated to each T-curve we
construct a surface, the T-filling, which is topologically the analog of the
quotient of the complexification of a real algebraic curve by the complex
conjugation. We use then the T-filling to prove a theorem for T-curves on
arbitrary ambient surfaces which is the analog of a theorem of Harnack for
algebraic curves, and to define the orientability and orientation of a T-curve,
the same way it is defined in the real algebraic setting.Comment: LaTeX2e, 33 pages, 18 figure
Generalised Veroneseans
In \cite{ThasHVM}, a characterization of the finite quadric Veronesean
by means of properties of the set of its tangent
spaces is proved. These tangent spaces form a {\em regular generalised dual
arc}. We prove an extension result for regular generalised dual arcs. To
motivate our research, we show how they are used to construct a large class of
secret sharing schemes
On k-sets of class [0, q/2-1, q/2, q/2+ 1, q] in a plane of even order q
Sets of class [0, q/2-1, q/2, q/2+1, q] in a plane of even order q ≥ 4 are classified and all possible sets are built. Some of the proofs require lemmas on more general classes of sets in planes of arbitrary order; namely, sets of types (0, n, q), (0, n - 1, n), (0, n - 1, n, q), and (0, n - 1, n, n + 1, q)
Abstract hyperovals, partial geometries, and transitive hyperovals
Includes bibliographical references.2015 Summer.A hyperoval is a (q+2)- arc of a projective plane π, of order q with q even. Let G denote the collineation group of π containing a hyperoval Ω. We say that Ω is transitive if for any pair of points x, y is an element of Ω, there exists a g is an element of G fixing Ω setwise such that xg = y. In1987, Billotti and Korchmaros proved that if 4||G|, then either Ω is the regular hyperoval in PG(2,q) for q=2 or 4 or q = 16 and |G||144. In 2005, Sonnino proved that if |G| = 144, then π is desarguesian and Ω is isomorphic to the Lunelli-Sce hyperoval. For our main result, we show that if G is the collineation group of a projective plane containing a transitivehyperoval with 4 ||G|, then |G| = 144 and Ω is isomorphic to the Lunelli-Sce hyperoval. We also show that if A(X) is an abstract hyperoval of order n ≡ 2(mod 4); then |Aut(A(X))| is odd. If A(X) is an abstract hyperoval of order n such that Aut(A(X)) contains two distinct involutions with |FixX(g)| and |FixX(ƒ)| ≥ 4. Then we show that FixX(g) ≠ FixX(ƒ). We also show that there is no hyperoval of order 12 admitting a group whose order is divisible by 11 or 13, by showing that there is no partial geometry pg(6, 10, 5) admitting a group of order 11 or of order 13. Finally, we were able to show that there is no hyperoval in a projective plane of order 12 with a dihedral subgroup of order 14, by showing that that there is no partial geometry pg(7, 12, 6) admitting a dihedral group of order 14. The latter results are achieved by studying abstract hyperovals and their symmetries
On Mathon's construction of maximal arcs in Desarguesian planes. II
In a recent paper [M], Mathon gives a new construction of maximal arcs which
generalizes the construction of Denniston. In relation to this construction,
Mathon asks the question of determining the largest degree of a non-Denniston
maximal arc arising from his new construction. In this paper, we give a nearly
complete answer to this problem. Specifically, we prove that when and
, the largest of a non-Denniston maximal arc of degree in
PG(2,2^m) generated by a {p,1}-map is (\floor {m/2} +1). This confirms our
conjecture in [FLX]. For {p,q}-maps, we prove that if and ,
then the largest of a non-Denniston maximal arc of degree in
PG(2,2^m) generated by a {p,q}-map is either \floor {m/2} +1 or \floor{m/2}
+2.Comment: 21 page
Maximally inflected real rational curves
We introduce and begin the topological study of real rational plane curves,
all of whose inflection points are real. The existence of such curves is a
corollary of results in the real Schubert calculus, and their study has
consequences for the important Shapiro and Shapiro conjecture in the real
Schubert calculus. We establish restrictions on the number of real nodes of
such curves and construct curves realizing the extreme numbers of real nodes.
These constructions imply the existence of real solutions to some problems in
the Schubert calculus. We conclude with a discussion of maximally inflected
curves of low degree.Comment: Revised with minor corrections. 37 pages with 106 .eps figures. Over
250 additional pictures on accompanying web page (See
http://www.math.umass.edu/~sottile/pages/inflected/index.html
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