177 research outputs found
Computing the characteristic numbers of the variety of nodal plane cubics in P3
AbstractIn this note we obtain, phrased in present day geometric and computational frameworks, the characteristic numbers of the family Unod of non-degenerate nodal plane cubics in P3, first obtained by Schubert in his Kalkül der abzählenden Geometrie. The main geometric contribution is a detailed study of a variety Xnod, which is a compactification of the family Unod, including the boundary components (degenerations) and a generalization to P3 of a formula of Zeuthen for nodal cubics in P2. The computations have been carried out with the Wiris boost WIT
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
Calabi–Yau threefolds and moduli of abelian surfaces I
We describe birational models and decide the rationality/unirationality of moduli spaces d (and levd) of (1, d)-polarized Abelian surfaces (with canonical level structure, respectively) for small values of d. The projective lines identified in the rational/unirational moduli spaces correspond to pencils of Abelian surfaces traced on nodal threefolds living naturally in the corresponding ambient projective spaces, and whose small resolutions are new Calabi–Yau threefolds with Euler characteristic zero
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
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