Point Corrections and Wall-Crossing Geometry on the Quintic del Pezzo Threefold

Abstract

Let $X=V_5$ be the quintic del Pezzo threefold. Along the negative zero-tilt-slope paths for rational-curve ideal classes, degree five is the first degree admitting a positive point correction. We determine the resulting degree-five outer good moduli space: it is reduced with exactly two irreducible components, a normal ten-dimensional principal branch and a smooth corrected branch $X\times\mathrm{Gr}(2,7)$. Their scheme-theoretic intersection is a normal determinantal ninefold. Along its singular locus, the principal branch has transverse rank-one $2\times 4$ determinantal singularities, while the intersection has the corresponding $2\times 3$ singularities. We construct an explicit small resolution of the principal branch as the projectivization of a rank-five extension bundle over the blow-up of the Altavilla--Petković--Rota Kuznetsov-component sixfold along its point locus. Uniform wall exclusion and point-correction bounds for degrees two through five establish the first correction threshold; the corrected degree-five factors are precisely the ideals of elliptic linear sections parametrized by $\mathrm{Gr}(2,7)$.