Coskun–Larson–Vogt conjecture on balanced normal bundles in Grassmannians

Establish that the normal bundle of a general nondegenerate rational curve of degree d in a Grassmannian G(k,n) is balanced except in the following cases: (i) 1 < d < min(k,n−k), or min(k,n−k) < d < max(k,n−k); (ii) writing d = kq₁ + r₁ with 0 ≤ r₁ < k and d = (n−k)q₂ + r₂ with 0 ≤ r₂ < n−k, one has d ≠ 1, r₁ ≠ 0, r₂ ≠ 0, and q₁ + q₂ ≤ (k−r₁)(n−k−r₂); or (iii) the characteristic is 2, k = 1 or k = n−1, and d is not congruent to 1 modulo n−2.

Background

The paper studies when the normal bundle of a general rational curve in a Grassmannian splits into line bundles whose degrees differ by at most one, a property called balancedness. The naive extension of the corresponding result for rational curves in projective space fails in Grassmannians because of explicitly identified degeneracy, tangent-bundle-splitting, and characteristic-2 phenomena.

Coskun–Larson–Vogt proposed that these phenomena account for all failures of balancedness. The paper proves the conjecture only for G(2,4) and G(2,6) in characteristic zero and for G(2,5) in arbitrary characteristic, leaving the general Grassmannian case unresolved.

References

Coskun-Larson-Vogt proposed that these are the only exceptions to the normal bundle being balanced and made the following conjecture.

The normal bundle of a general nondegenerate rational curve of degree $d$ in a Grassmannian $G(k,n)$ is balanced except for the following cases: \begin{enumerate}[(i)] \item $1 < d < \min(k,n-k)$ or $\min(k,n-k) < d < \max(k, n - k)$, \item If we write $d = k q_1 + r_1$ with $0 \leq r_1 < k$ and $d = (n - k) q_2 + r_2$ with $0 \leq r_2 < n - k$ then $d \neq 1$, $r_1 \neq 0$, $r_2 \neq 0$, and $q_1 + q_2 \leq (k - r_1)(n - k - r_2)$. \item The characteristic is 2, $k=1$ or $k = n-1$, and $d \not \equiv 1 \bmod{n-2}$. \end{enumerate}

— Balancedness of Normal Bundles of Rational Curves in Grassmannians  (2609.19381 - Cao, 16 Sep 2026) in Section 1, Conjecture 1.1 (labeled Conjecture \ref{conjecture})