Nash blowing-up¶
Replace a singular variety by the closure of the graph of its smooth-point tangent-space map, retaining each singular point together with the limiting tangent spaces approached nearby.
Core Idea¶
The Nash blowing-up of a pure-dimensional variety is the closure of the graph of the map sending each smooth point to its tangent space, equipped with the projection back to the variety. The smooth locus records a unique tangent plane at every regular point; closing its graph over singular points adds precisely the tangent planes that arise as limits along nearby smooth points and produces a proper birational modification.
Its autonomous residual is the tangent-graph closure and its birational projection, not generic blow-up terminology, the ordinary blow-up at one arbitrarily chosen ideal, or a complete resolution of every singularity.
Scope of Application¶
Nash blowing-up applies when the analyst can specify a reduced pure-dimensional algebraic variety, its smooth locus, and the Grassmann bundle of tangent planes in a smooth ambient embedding or an equivalent intrinsic construction and establish that the modified space is the graph closure of the smooth-locus Gauss map and its morphism to the original variety restricts to an isomorphism over the smooth locus. The entry states a descriptive algebraic-geometric construction. It does not present software commands, a computational attack surface, or a laboratory procedure.
Clarity¶
A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because Nash blow-up, Nash modification, and Nash blowing-up are near-synonyms, while higher Nash blow-ups and Nash's arc-space problem are related but distinct objects. The disciplined statement is that the object counts as Nash blowing-up exactly when the modified space is the graph closure of the smooth-locus Gauss map and its morphism to the original variety restricts to an isomorphism over the smooth locus
Manages Complexity¶
The abstraction compresses classical and higher Nash blow-ups, algebraic and analytic varieties, intrinsic and embedded presentations, curve and higher-dimensional cases, Jacobian descriptions, and characteristic-dependent behavior into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.
Compression can hide assumptions. A responsible use therefore declares dimension, smooth locus, embedding, Grassmannian, tangent limit, graph closure, exceptional fiber, Jacobian ideal, properness, birationality, iteration, and characteristic and returns to the full diagnostic whenever a convention or boundary case changes.
Abstract Reasoning¶
- Type the carrier. Establish a reduced pure-dimensional algebraic variety, its smooth locus, and the Grassmann bundle of tangent planes in a smooth ambient embedding or an equivalent intrinsic construction and reject examples from a different problem. 2. Lock the rule. Express that the modified space is the graph closure of the smooth-locus Gauss map and its morphism to the original variety restricts to an isomorphism over the smooth locus independently of one notation or implementation.
Knowledge Transfer¶
Transfer within algebraic geometry is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from For an embedded variety, map every nonsingular point to the pair consisting of that point and its tangent plane in the appropriate Grassmannian, then take the closure of that graph. to For a complete-intersection presentation, the Nash blow-up can be related to a blow-up governed by maximal minors of a Jacobian matrix. demonstrates that continuity.
Relationships to Other Abstractions¶
Current abstraction Nash blowing-up Domain-specific
Parents (1) — more general patterns this builds on
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Nash blowing-up is a kind of Transformation Prime
The proposed strict upward parent is
prime:transformation.
Hierarchy path (1) — routes to 1 parentless root
- Nash blowing-up → Transformation → Function (Mapping)
Neighborhood in Abstraction Space¶
Nash blowing-up sits in a moderately populated region (54th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Varieties, Morphisms & Birational Geometry (12 abstractions)
Nearest neighbors
- Severi–Brauer variety — 0.88
- Spherical variety — 0.88
- Stratifold — 0.88
- Smooth functor — 0.87
- Geometric quotient — 0.87
Computed from structural-signature embeddings · 2026-09-08