Some loci of rational cubic fourfolds

By M. Bolognesi, Francesco Russo, Giovanni Staglianò. Mathematische Annalen, 2015

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In this paper we investigate the divisor $${\mathcal {C}}_{14}$$C14 inside the moduli space of smooth cubic hypersurfaces in $${\mathbb {P}}^5$$P5, whose general element is a smooth cubic containing a smooth quartic rational normal scroll. By showing that all degenerations of quartic scrolls in $${\mathbb {P}}^5$$P5 contained in a smooth cubic hypersurface are surfaces with one apparent double point, we prove that every cubic hypersurface contained in $${\mathcal {C}}_{14}$$C14 is rational. Combining our proof with the Hodge theoretic definition of $${\mathcal {C}}_{14}$$C14, we deduce that on a smooth cubic fourfold every class $$T\in {\text {H}}^{2,2}(X,\mathbb {Z})$$T∈H2,2(X,Z) with $$T^2=10$$T2=10 and $$T\cdot h^2=4$$T·h2=4 is represented by a (possibly reducible) surface of degree four which has one apparent double point. As an application of our results and of the construction of some explicit examples, we also prove that the Pfaffian locus is not open in $${\mathcal {C}}_{14}$$C14.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4a2aa786ca065c346983433892682180827f5154

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In Geometry and Topology

  1. Brendan Hassett
  2. D. Smyth
  3. Donghoon Hyeon
  4. Franccois Charles
  5. Giovanni Mongardi
  6. Brian Lehmann
  7. Chenyang Xu
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In Mathematical Physics

  1. Brendan Hassett
  2. Donghoon Hyeon
  3. D. Smyth
  4. Giovanni Mongardi
  5. Brian Lehmann
  6. Chenyang Xu
  7. Mihai Fulger

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