Irreducibility and birational geometry of the main rank-three degeneracy component

Determine, for triples (k,n,r) with r≥4, when the main irreducible component D_3(Φ_E)^{\mathrm{main}} is irreducible for general E, and determine when the component of the incidence model \widetilde D_3(E) dominating it is smooth, rational, or stably rational.

Background

For codimension r≥4, the minimal-envelope conic locus is controlled by the rank-three degeneracy locus D_3(Φ_E) associated with the map Φ_E:P→E∨. The paper proves that, when nonempty, this degeneracy locus has the expected codimension and that the exact-rank-three stratum is related birationally to the corresponding minimal-envelope conics, subject to density assumptions.

The paper does not determine the global irreducibility or birational geometry of the relevant component in general. In particular, the behavior of the dominating component of the incidence resolution \widetilde D_3(E) remains unresolved beyond the low-codimension range.

References

For which triples (k,n,r) with r\ge4 is D_3(\Phi_E){\mathrm{main} irreducible for general E? When is the dominating component of \widetilde D_3(E) smooth, rational, or stably rational?

— Moduli of Conics on General Plucker Linear Sections of Grassmannians  (2609.04727 - Fu, 4 Sep 2026) in Section Further questions, first Question

Can the Thom--Porteous class of D_3(\Phi_E) and the canonical formula~eq:canonical be combined to classify the Fano, Calabi--Yau, and general-type cases of expected dimension at most three?

eq:canonical:

KSE=[π∗KB⊗π∗(det⁡P)−3⊗(det⁡R∨)r−6⊗(det⁡E∨)3]∣SE.K_{S_E} = \left[ \pi^*K_B\otimes\pi^*(\det P)^{-3} \otimes(\det\mathcal R^\vee)^{r-6} \otimes(\det E^\vee)^3 \right]\Big|_{S_E}.

— Moduli of Conics on General Plucker Linear Sections of Grassmannians  (2609.04727 - Fu, 4 Sep 2026) in Section Further questions, second Question