A variety or scheme that is not normal, i.e., at some point one of its local rings is not integrally closed in its total ring of fractions. Nonnormality produces branch-like phenomena, failure of integral closure, and can manifest as cusps, pinch points, or more subtle failures of being integrally closed in codimension one.

Principle

Normality organizes geometric coherency by requiring integral closure of local rings; equivalently (by Serre) normality is characterized by regularity in codimension one (R_1) together with the depth condition S_2. Failure of integrality in local rings leads to separated branches in the normalization and to obstructions in divisor theory.

Demonstration

Standard example: the cusp Spec k[t^2,t^3] is not normal because t is integral over k[t^2,t^3] but does not lie in the subring; the normalization is Spec k[t] which separates the parametrizing line. Another concrete case is a variety with a pinch point where the normalization map replaces a nonintegrally closed local ring by its integral closure, splitting or smoothing branches.

Misapplication

Assuming 'nonnormal' is equivalent to 'singular' or to 'reducible'; a scheme may be singular yet normal in dimension greater than one, or reduced yet nonnormal. Confusing nonreducedness with nonnormality is common: a reduced scheme can still fail to be integrally closed.

Consequence

Nonnormality affects divisor class groups, Weil vs Cartier divisor distinctions, and the behavior of sheaves: reflexive sheaves and dualizing complexes behave differently on nonnormal loci. The normalization map modifies topology and can separate branches, altering numerical and categorical invariants.

Reversal

A normal variety has integrally closed local rings everywhere; its codimension-one local rings are regular and divisors behave predictably (Weil divisors correspond more closely to Cartier data). Normality removes branch-like singular behavior coming from integral closure failure.

Boundary

The term is most meaningful for integral or reduced schemes of finite type; for reducible schemes one examines normality componentwise. Nonnoetherian rings and some nonreduced examples require extra care: integral closure can be subtle, and the usual equivalences (e.g., Serre's criterion) require Noetherian hypotheses.

Semantic Tension

Nearby notions: 'regular', 'normal', and 'reduced' are distinct. Users often conflate 'nonnormal' with 'singular' or with 'nonreduced'; the tension lies in that normality is an integrality condition while smoothness/regularity are differential conditions and reducedness concerns nilpotents.

Synthesis

A Nonnormal Variety is one whose local algebra fails integral closure: locally some element integral over the ring lies outside it, producing branch-like behavior that the normalization corrects. This failure has concrete geometric consequences for divisors, normalization maps, and the splitting of branches in the geometry.