The subset (often closed) of a scheme or variety consisting of all points that are singular; scheme-theoretically it may be defined by the vanishing of minors of the Jacobian of defining equations.

Principle

The singular locus collects points where local regularity fails uniformly; algebraically it is cut out by the ideal generated by the appropriate minors of the Jacobian (or by the Fitting ideals of the sheaf of differentials), making it a determinantal condition and therefore typically closed.

Demonstration

For a hypersurface defined by f in affine space, the singular locus is defined by the simultaneous vanishing of f and all its partial derivatives; for example the curve f=y^2-x^3 has singular locus equal to the origin.

Misapplication

Treating the singular locus as merely the complement of a smooth open set without scheme structure, or assuming it is always finite — some varieties have positive-dimensional singular loci or embedded components.

Consequence

Knowing the singular locus allows stratification into smooth and singular strata, guides resolution procedures (blow-ups centered along components), and informs invariants like arithmetic genus or intersection multiplicities.

Reversal

The regular (smooth) locus, the open subset where the variety is regular; its complement is the singular locus.

Boundary

Applies scheme-theoretically; behavior under base change, normalization, or completion must be considered. Excludes analytic or differential singular supports that are not visible algebraically.

Semantic Tension

Confused with 'singular support' or 'critical locus' in other contexts; the singular locus is a geometric/schematic set of points where regularity fails, whereas 'critical' may refer to a map's differential vanishing or to microlocal supports.

Synthesis

The singular locus is the algebraic subset whose points fail local regularity, typically defined by determinantal conditions on the Jacobian and serving as the canonical locus for resolution and stratification operations.