Obstructions to the deformations of curves to other hypersurfaces

2011/10/02 by Bin Wang, Wang, Bin
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG

paper · pdf · doi:10.48550/arxiv.1110.0184

arxiv created 2011/12/28 · arxiv updated 2011/12/30

Abstract

We are interested in obstructions to the FIRST order deformation of a pair of a smooth hypersurface f₀ and a smooth curve C₀ contained in f₀. In the first half of the paper, we give necessary conditions for the pair to deform in the first order. In particular, for a rational curve C₀, this necessary condition is H¹(N_C0f₀(1))=0. In the second half, we apply the necessary conditions from the first half of the paper to study the geometry of smooth curves in hypersurfaces. The main application is for the case where C₀ is a rational curve.

Related

• Genus of curves in generic hypersurfaces
• Obstructions to deforming maps from curves to surfaces
• Deformation of singular curves on surfaces
• Infinitesimal deformations of complements of plumbings of rational curves
• Obstructions to deforming space curves lying on a smooth cubic surface