99 research outputs found
Plane quartics with at least 8 hyperinflection points
A recent result shows that a general smooth plane quartic can be recovered
from its 24 inflection lines and a single inflection point. Nevertheless, the
question whether or not a smooth plane curve of degree at least 4 is determined
by its inflection lines is still open. Over a field of characteristic 0, we
show that it is possible to reconstruct any smooth plane quartic with at least
8 hyperinflection points by its inflection lines. Our methods apply also in
positive characteristic, where we show a similar result, with two exceptions in
characteristic 13
Recovering plane curves of low degree from their inflection lines and inflection points
In this paper we consider the following problem: is it possible to recover a
smooth plane curve of degree at least three from its inflection lines? We
answer positively to the posed question for a general smooth plane quartic
curve, making the additional assumption that also one inflection point is
given, and for any smooth plane cubic curve.Comment: 24 page
Reconstructing general plane quartics from their inflection lines
Let be a general plane quartic and let denote the
configuration of inflection lines of . We show that if is any plane
quartic with the same configuration of inflection lines , then the
quartics and coincide.Comment: 21 pages, to appear in Transactions of the American Mathematical
Societ
The infinite random simplicial complex
We study the Fraisse limit of the class of all finite simplicial complexes.
Whilst the natural model-theoretic setting for this class uses an infinite
language, a range of results associated with Fraisse limits of structures for
finite languages carry across to this important example. We introduce the
notion of a local class, with the class of finite simplicial complexes as an
archetypal example, and in this general context prove the existence of a 0-1
law and other basic model-theoretic results. Constraining to the case where all
relations are symmetric, we show that every direct limit of finite groups, and
every metrizable profinite group, appears as a subgroup of the automorphism
group of the Fraisse limit. Finally, for the specific case of simplicial
complexes, we show that the geometric realisation is topologically surprisingly
simple: despite the combinatorial complexity of the Fraisse limit, its
geometric realisation is homeomorphic to the infinite simplex.Comment: 33 page
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