Linear subspaces of hypersurfaces

2019/03/06 by Beheshti, Roya, Riedl, Eric · 4 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1903.02481

Abstract

Let X be an arbitrary smooth hypersurface in ℂ ℙⁿ of degree d. We prove the de Jong-Debarre Conjecture for n ≥ 2d-4: the space of lines in X has dimension 2n-d-3. We also prove an analogous result for k-planes: if n ≥ 2 \binomd+k-1k + k, then the space of k-planes on X will be irreducible of the expected dimension. As applications, we prove that an arbitrary smooth hypersurface satisfying n ≥ 2^d! is unirational, and we prove that the space of degree e curves on X will be irreducible of the expected dimension provided that d ≤ (e+n)/(e+1).

Cited by

• Separable rational connectedness and k-plane sections of hypersurfaces
• Unirationality of hypersurfaces via highly tangent lines
• Isotropy and completeness indices of multilinear maps
• Profiles, linear spaces, and unirationality of complete intersections

Related

• On the Debarre-de Jong and Beheshti-Starr conjectures on hypersurfaces with too many lines
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• Unirationality of hypersurfaces via highly tangent lines
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• On a Conjecture on the Variety of Lines on a Fano Complete Intersection