The subset of a parameter (moduli) space where the dimension of a given cohomology group of fibers increases discontinuously compared to the general fiber; places where cohomology 'jumps' indicating special geometric or algebraic features.

Principle

Upper semicontinuity of cohomology dimensions under reasonable hypotheses ensures that loci of larger dimension are closed (or constructible) and that jumps reflect failure of generic vanishing, flatness degenerations, or emergence of new global sections or extension classes.

Demonstration

In a flat family of coherent sheaves on a scheme, points where H^i of the fiber has dimension strictly bigger than at a nearby general point form the cohomology jumping locus. For example, line bundles on curves may acquire extra H^0 at special points of a moduli parameter, producing Brill‑Noether type loci.

Misapplication

Inferring severe singularity or nonflatness of the entire family only from a cohomology jump at isolated parameters ignores possibilities like special fibers with extra automorphisms, torsion, or accidental global sections; the jump must be interpreted with auxiliary data.

Consequence

Identifying jumping loci yields stratifications of parameter spaces, informs stability conditions, controls deformation theory, and pinpoints where additional geometric structures or extension classes appear; it guides constructing walls and chambers in moduli problems.

Reversal

Away from the jumping locus, cohomology dimensions are minimal and behave predictably (often constant in families satisfying flatness), so generic points exhibit no jump and simpler geometric behavior prevails.

Boundary

This concept requires a parameterized family with well‑defined fiberwise cohomology (schemes, complex varieties, sheaves) and does not apply to isolated cohomology invariants without a parameter space. It excludes codimension‑zero trivial variations where dimension is constant.

Semantic Tension

The term competes with 'specialization phenomena' more broadly: jumping loci single out discrete increases in cohomology dimension, while other specializations may change qualitative geometry without a straightforward cohomology jump.

Synthesis

A Cohomology Jumping Locus is the geometric place in a parameter space where cohomology gains rank, serving as an indicator of special fibers, obstructed deformations, or emergent global sections; it organizes loci of exceptional behavior and guides refined moduli analysis.