Dimca–Sticlaru conjecture

The Dimca–Sticlaru conjecture asserts that every rational cuspidal plane curve is free or nearly free. Earlier work established this only in substantial special cases, while a 2023 survey still described the general conjecture as open.

Known results

• Even degree, prime-power degree, or abelian complement fundamental group: the conjecture holds (Dimca–Sticlaru, 2015).
• Many odd degrees, including all degrees at most $34$, were settled (Dimca–Sticlaru, 2018).
• For odd degree $d=2d'+1$, $\operatorname{mdr}(f)\le d'$; equality implies freeness or near-freeness (Dimca–Sticlaru, 2018).
• The full conjecture was still reported open in 2023.

September 2026 claimed proof

On September 21, 2026, Xiping Zhang's preprint On Free and Nearly Free Conjecture of Rational Unibranched Projective Plane Curves claimed the inequality $\chi(U)\ge\nu(C)$ and applied it to the rational unibranched case, thereby claiming the conjecture. The preprint's claim is unverified.

Current status (as of September 2026): The conjecture has a preprint claiming a complete proof for the stated reduced irreducible complex-curve setting, but that proof remains unverified; previously only partial cases were settled.