[Paper Review] Obstructions to deforming maps from curves to surfaces

Takeo Nishinou|arXiv (Cornell University)|Jan 31, 2019

Algebraic Geometry and Number Theory10 references4 citations

TL;DR

This paper extends the classical notion of semiregularity to maps from curves to surfaces, enabling control over deformations of singular curves—particularly nodal and non-reduced curves—by introducing a local, map-based semiregularity condition. The key result shows that if a map from a reduced curve to a surface is semiregular in this extended sense, its deformations are unobstructed, allowing geometrically controlled constructions of curves with many nodes on surfaces of general type.

ABSTRACT

This paper studies the obstructions to deforming a map from a complex variety to another variety which is an immersion of codimension one. We extend the classical notion of semiregularity of subvarieties to maps between varieties, and show that it largely extends the applicability. Then we apply the main result to several situations. First, we give deformations of non-reduced curves on surfaces in a geometrically controlled way. Also, we give a simple but effective criterion for the vanishing of the obstructions to equisingular deformations of nodal curves on surfaces. Finally, we construct nodal curves with very small geometric genus on surfaces of general type.

Motivation & Objective

• To extend the classical semiregularity condition from subvarieties to maps between varieties, particularly from curves to surfaces.
• To provide a local, map-based criterion for unobstructed deformations of singular curves on surfaces.
• To construct nodal curves with very large numbers of nodes on surfaces of general type, where classical deformation theory fails.
• To control geometric properties of deformed curves by restricting the domain of the map, enabling equisingular and non-reduced curve deformations.

Proposed method

• Introduce a new notion of semiregularity for maps φ: V → X, where V is a smooth variety (dim X > 2) or a reduced curve (dim X = 2), and X is a smooth surface.
• Define semiregularity of φ via the vanishing of obstructions in the cohomology of the normal sheaf Nφ, using the exact sequence involving Nφ(−P) and the skyscraper sheaves at nodes.
• Use Serre duality and the Leray spectral sequence to relate the cohomology of Nφ to global sections of the canonical bundle on the domain curves.
• Establish a criterion for semiregularity of maps whose image is a nodal curve: φ is semiregular iff, for each node pi, there exists a first-order deformation of the image curve that smoothes pi but not the other nodes.
• Apply the criterion to reducible curves by gluing semiregular components and using the compatibility of normal sheaves under restriction and pushforward.
• Construct deformations of nodal curves by iteratively adding sections of the normal sheaf that smooth individual nodes, ensuring the deformation extends algebraically via Theorem 24.

Experimental results

Research questions

• RQ1 Can the classical semiregularity condition be generalized from subvarieties to maps from curves to surfaces to allow control over deformation properties of singular curves?
• RQ2 What local criterion ensures that a map from a reduced curve to a surface is unobstructed in its deformation space?
• RQ3 Can nodal curves with very large numbers of nodes be constructed on surfaces of general type using a map-based semiregularity condition?
• RQ4 How can non-reduced curves be deformed in a geometrically controlled way using the extended semiregularity framework?
• RQ5 What conditions ensure that a deformation of a nodal curve preserves equisingularity or controls the number of nodes?

Key findings

• The map φ: V → X is unobstructed if it is semiregular in the extended sense, meaning all first-order deformations extend to higher-order deformations.
• When the canonical bundle of X is trivial, any immersion from a reduced curve is semiregular, enabling unobstructed deformations of non-reduced curves.
• For nodal curves, φ is semiregular if and only if for each node pi, there exists a first-order deformation of the image curve that smooths pi but not the other nodes.
• A criterion in terms of linear systems shows that φ is semiregular if, for each node pi, there exists an effective divisor algebraically equivalent to φ(C) that avoids pi but passes through all other nodes.
• The geometric genus of the deformed curve C(n) is bounded by ng(C1) + (n−1)KX·L, and for sufficiently ample L, the number of nodes can be made to satisfy Aδ(C) > g(C)².
• The surface X admits a family of nodal curves satisfying the semiregularity condition, demonstrating the existence of curves with very large numbers of nodes on general type surfaces.