The Unenriched Count

Enumerative geometry counts geometric objects satisfying specified conditions — lines on a surface, curves through points, intersections of varieties. Over the complex numbers, these counts are integers. Over the real numbers, the counts depend on the configuration and lack the cleanness of the complex answer.

Quadratic enrichment, developed through A¹-homotopy theory, replaces integer counts with elements of the Grothendieck-Witt group — quadratic forms that encode not just “how many” but “with what orientation.” The enriched count carries more information than the classical one and is supposed to be more natural.

arXiv:2603.16315 computes the quadratically enriched count of lines on smooth del Pezzo surfaces of degree 2 and 4. The result: the enrichment is trivial. The Chow-Witt group relevant to the count is isomorphic to the ordinary Chow group. The quadratic form collapses to a multiple of the hyperbolic element. The enrichment that was supposed to add structure adds nothing.

The technical reason is that the vector bundles involved are not relatively orientable over the relevant Grassmannians, and the resulting Chow-Witt groups lose their quadratic refinement. The enrichment machinery works — it just outputs the same answer the unenriched theory already gave.

This is a structural negative result: some enumerative problems are genuinely enrichable (they carry quadratic information invisible to classical counts) and some are not (the enrichment collapses). The distinction is not obvious in advance — it depends on the orientability of specific bundles in the specific geometric setup. The enrichment’s power has a boundary, and this computation locates part of that boundary.