Monomial embeddings of the Klein curve

AbstractThe Klein curve is defined by the smooth plane model X3Y+Y3Z+Z3X=0. We give all embeddings in higher dimension with a linear action of the automorphism group. The curve has 24 flexpoints, i.e. points where the tangent intersects with multiplicity three. For even characteristic, the embeddings yield interesting configurations of the flexpoints and good linear codes

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

Degenerations of elliptic curves and cusp singularities

We give more or less explicit equations for all two-dimensional cusp singularities of embedding dimension at least 4. They are closely related to Felix Klein's equations for universal curves with level n structure. The main technical result is a description of the versal deformation of an n-gon in $P^{n-1}$. The final section contains the equations for smoothings of simple elliptic singularities (of multiplicity at most 9).Comment: Plain Te

Asymptotic enumeration and limit laws for graphs of fixed genus

It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface S_g of genus g grows asymptotically like $c^{(g)}n^{5(g-1)/2-1}\gamma^n n!$ where $c^{(g)}>0$, and $\gamma \approx 27.23$ is the exponential growth rate of planar graphs. This generalizes the result for the planar case g=0, obtained by Gimenez and Noy. An analogous result for non-orientable surfaces is obtained. In addition, it is proved that several parameters of interest behave asymptotically as in the planar case. It follows, in particular, that a random graph embeddable in S_g has a unique 2-connected component of linear size with high probability