Parameter spaces
Hilbert schemes
A Hilbert scheme packages algebraic subvarieties, such as curves in projective space, into a geometric space. Its local geometry records how those curves can vary.
Algebraic Geometry · Hilbert Schemes · Deformation Theory
Hirokazu Nasu
I study how algebraic curves move inside projective varieties, and why some apparently possible infinitesimal motions are obstructed. These phenomena are reflected in singular and non-reduced structures of Hilbert schemes.
Professor, Department of Mathematical Sciences, School of Science, Tokai University, Japan
Research at a glance
My work lies at the intersection of deformation theory, the geometry of curves, and moduli-type parameter spaces.
Local geometry
Deformations and obstructions
Infinitesimal deformations describe first-order motions of a curve. Obstructions detect when such motions cannot be extended to genuine families.
Ambient geometry
Curves on special varieties
I investigate curves lying on cubic and del Pezzo surfaces, K3 surfaces, Fano varieties, and complete intersections of quadrics.
Why can a curve fail to move?
A central question in deformation theory is whether a first-order motion can be continued. Failure produces an obstruction and often signals singular behavior in the Hilbert scheme.
A curve CStart with an embedded algebraic curve in a projective variety.
First-order motionStudy an infinitesimal deformation, represented by a tangent direction in the Hilbert scheme.
Can it extend?Ask whether the infinitesimal motion comes from an actual one-parameter family of curves.
ObstructionIf extension fails, the obstruction contributes to singular or non-reduced structure of the Hilbert scheme.
Selected publications
A selection tracing the development of my work on obstructions and Hilbert schemes.
Obstructions to deforming space curves and non-reduced components of the Hilbert scheme
Obstructions to deforming curves on a 3-fold, I: A generalization of Mumford's example and an application to Hom schemes
Obstructions to deforming curves on a 3-fold, III: Deformations of curves lying on a K3 surface
Obstructions to deforming space curves lying on a smooth cubic surface
Obstructions to deforming space curves lying on a del Pezzo surface
Current direction
Complete intersections of quadrics
Recent work develops deformation and obstruction techniques for curves lying on complete intersections of quadrics, including genus-5 K3 surfaces, and related questions about non-reduced components of Hilbert schemes.
Recent lecture series
Deformation Theory of Embedded Curves and Its Applications, Korea University, September 2026: three lectures and one research talk on obstructions to deforming curves lying on a complete intersection of quadrics.
Deformation Theory of Embedded Curves and Its Applications, Korea University, September 2026: three lectures and one research talk on obstructions to deforming curves lying on a complete intersection of quadrics.
ACADEMIC BACKGROUND
Research journey
I received my Ph.D. from Kyoto University in 2005, where I studied at the Research Institute for Mathematical Sciences (RIMS) under the supervision of Prof. Shigeru Mukai . After completing my Ph.D., I spent three years at RIMS and six months at the University of Oslo as a postdoctoral researcher. During my stay in Oslo, I was supported by the Japan–Norway Researcher Mobility Programme (FY2008) , administered by the Research Council of Norway (RCN) in cooperation with the Japan Society for the Promotion of Science (JSPS), and worked in collaboration with Prof. Kristian Ranestad and Prof. Jan Oddvar Kleppe .
Contact
Department of Mathematical Sciences, School of Science, Tokai University
4-1-1 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan