Hirokazu Nasu
Research overview

Hilbert schemes and deformation theory

How do algebraic curves move? When does an infinitesimal motion extend to an actual family, and what can failure tell us about the geometry of the parameter space?

Curves as points of a geometric space

Suppose that a curve C is embedded in projective space, or more generally in a fixed projective variety. Rather than studying one curve at a time, algebraic geometry places curves with the same numerical data into a parameter space called a Hilbert scheme. A point of the Hilbert scheme represents an embedded algebraic object, while paths and local directions in that space encode possible deformations.

This turns a question about “moving a curve” into a geometric question about the local structure of the Hilbert scheme. If the Hilbert scheme is smooth at the point corresponding to C, infinitesimal behavior matches the expected local geometry. If it is singular or non-reduced, more subtle phenomena occur.

First-order deformations and obstructions

An infinitesimal deformation is a first-order approximation to a genuine family of curves. Such deformations form the tangent space to the Hilbert scheme at the point [C]. But not every tangent direction necessarily extends to higher order.

An obstruction measures this failure. From this viewpoint, obstructions are not merely technical difficulties: they explain why the Hilbert scheme can have singularities, unexpected tangent dimensions, or non-reduced components.

Embedded curveA curve C lies on a surface or threefold inside projective space.
Infinitesimal deformationA first-order motion gives a tangent vector to the Hilbert scheme.
Higher-order extensionOne tests whether the motion continues to an actual family.
Obstruction theoryFailure is detected by obstruction classes and reflected in the local scheme structure.

The role of the ambient surface or threefold

Much of my work exploits the geometry of a surface or threefold containing the curve. The ambient variety can make deformation calculations more explicit and can reveal the geometric source of an obstruction.

Classical starting point

Cubic surfaces

Mumford's example shows that the Hilbert scheme of space curves can have a non-reduced component. My early work generalized and developed obstruction calculations for curves on cubic surfaces.

K3 and Fano geometry

K3 surfaces and Fano varieties

Curves on K3 surfaces and related Fano varieties provide a rich setting in which deformation theory interacts with the geometry of the ambient variety.

Recent direction

Complete intersections of quadrics

Current work studies curves on complete intersections of quadrics, including genus-5 K3 surfaces, and the resulting obstruction phenomena in Hilbert schemes.

A continuing theme

Across these projects, the guiding problem is to relate three layers of geometry: the geometry of the curve, the geometry of the ambient surface or higher-dimensional variety, and the local scheme structure of the Hilbert scheme. This approach has produced examples and criteria for obstructed curves, singular points, and non-reduced components of Hilbert schemes.

For specialists. My research uses normal bundles, infinitesimal deformations, primary obstructions, Hilbert-flag schemes, and geometric constructions on cubic and del Pezzo surfaces, K3 surfaces, Fano varieties, and related embedded varieties.