Some remarks on varieties with degenerate Gauss image

We consider projective varieties with degenerate Gauss image whose focal hypersurfaces are non-reduced schemes. Examples of this situation are provided by the secant varieties of Severi and Scorza varieties. The Severi varieties are moreover characterized by a uniqueness property.Comment: 9 pages, to be published in Pacific Journal of Mathematic

Stable cohomology of spaces of non-singular hypersurfaces

We prove that the rational cohomology of the space of non-singular complex homogeneous polynomials of degree d in a fixed number of variables stabilizes to the cohomology of the general linear group for d sufficiently large.Comment: 11 pages; v3: stabilization range made explicit, proof of Lemma 3 corrected and expande

The Origin and Shaping of Planetary Nebulae: Putting the Binary Hypothesis to the Test

Planetary nebulae (PNe) are circumstellar gas ejected during an intense mass-losing phase in the the lives of asymptotic giant branch stars. PNe have a stunning variety of shapes, most of which are not spherically symmetric. The debate over what makes and shapes the circumstellar gas of these evolved, intermediate mass stars has raged for two decades. Today the community is reaching a consensus that single stars cannot trivially manufacture PNe and impart to them non spherical shapes and that a binary companion, possibly even a sub-stellar one, might be needed in a majority of cases. This theoretical conjecture has however not been tested observationally. In this review we discuss the problem both from the theoretical and observational standpoints, explaining the obstacles that stand in the way of a clean observational test and ways to ameliorate the situation. We also discuss indirect tests of this hypothesis and its implications for stellar and galactic astrophysics.Comment: 28 pages of text. 4 tables 9 figures. Accepted by PASP Review

Binary central stars of planetary nebula

Only a handful of binary central stars of planetary nebulae (PNe) are known today, due to the difficulty of detecting their companions. Preliminary results from radial velocity surveys, however, seem to indicate that binarity plays a fundamental,rather than marginal role in the evolution of PNe and that the close binary fraction might be much larger than the currently known value of 10-15%. In this review, we list all the known binary central stars, giving an updated census of their numbers and selected characteristics. A review is also given of the techniques used to detect binaries as well as selected characteristics of related stellar classes which might provide constraints (or additional puzzles) to the theory of PN evolution. Finally, we will formulate the conjecture that all PNe derive from binary interactions and suggest that this is not inconsistent with our current knowledge.Comment: 8 pages, no figures. Review presented at IAU Symposium 234 on Planetary Nebulae. Proceedings in pres

Cohomology of the second Voronoi compactification of A_4

In this paper we compute the cohomology groups of the second Voronoi compactification of the moduli space of abelian fourfolds in all degrees with the exception of the middle degree 10. We also compute the cohomology groups of the perfect cone compactification in degree < 10. The main tool is the investigation of the strata of the compactification corresponding to semi-abelic varieties with constant torus rank.Comment: v2: 41 pages, mostly expository change

On projective varieties of dimension n+k covered by k-spaces

We study families of linear spaces in projective space whose union is a proper subvariety X of the expected dimension. We establish relations between configurations of focal points and existence or non-existence of a fixed tangent space to X along a general element of the family. We apply our results to the classification of ruled 3-dimensional varieties.Comment: To be published in Illinois Journal of Mathematic