165 research outputs found
Equivariant Degenerations of Plane Curve Orbits
In a series of papers, Aluffi and Faber computed the degree of the
orbit closure of an arbitrary plane curve. We attempt to generalize this to the
equivariant setting by studying how orbits degenerate under some natural
specializations, yielding a fairly complete picture in the case of plane
quartics.Comment: 33 pages, comments welcom
Twenty-Seven Questions about the Cubic Surface
We present a collection of research questions on cubic surfaces in 3-space.
These questions inspired a collection of papers to be published in a special
issue of the journal Le Matematiche. This article serves as the introduction to
that issue. The number of questions is meant to match the number of lines on a
cubic surface. We end with a list of problems that are open.Comment: 13 page
Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds
We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau
3-folds starting with (almost) any deformation family of smooth weak Fano
3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau
3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. We
pay particular attention to a subclass of weak Fano 3-folds that we call
semi-Fano 3-folds. Semi-Fano 3-folds satisfy stronger cohomology vanishing
theorems and enjoy certain topological properties not satisfied by general weak
Fano 3-folds, but are far more numerous than genuine Fano 3-folds. Also, unlike
Fanos they often contain P^1s with normal bundle O(-1) + O(-1), giving rise to
compact rigid holomorphic curves in the associated ACyl Calabi-Yau 3-folds.
We introduce some general methods to compute the basic topological invariants
of ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds, and study a
small number of representative examples in detail. Similar methods allow the
computation of the topology in many other examples.
All the features of the ACyl Calabi-Yau 3-folds studied here find application
in arXiv:1207.4470 where we construct many new compact G_2-manifolds using
Kovalev's twisted connected sum construction. ACyl Calabi-Yau 3-folds
constructed from semi-Fano 3-folds are particularly well-adapted for this
purpose.Comment: 107 pages, 1 figure. v3: minor corrections, changed formattin
Twenty-seven questions about the cubic surface
We present a collection of research questions on cubic surfaces in 3-space. These questions inspired the present collection of papers. This article serves as the introduction to the issue. The number of questions is meant to match the number of lines on a cubic surface. We end with a list of problems that are open
Algebraic boundaries of Hilbert™s SOS cones
AbstractWe study the geometry underlying the difference between non-negative polynomials and sums of squares (SOS). The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether–Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized, respectively, by the Severi variety of plane rational sextics and by the variety of quartic symmetroids.GB, JH and BS were also supported
by the US National Science Foundation.This is the accepted manuscript. The final version is available from CUP at http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8764762&fileId=S0010437X12000437
Abelian surfaces with two plane cubic curve fibrations and Calabi-Yau threefolds
(1,d)-polarized abelian surfaces in P^(d-1) with two plane cubic curve
fibrations lie in two elliptic P^2-scrolls. The union of these scrolls form a
reducible Calabi-Yau 3-fold. In this paper we show that this occurs when d<10
and analyse the family of such surfaces and 3-folds in detail when d=6. In
particular, the reducible Calabi-Yau 3-folds deform in that case to irreducible
ones with non-normal singularities.Comment: 41 page
- …