SHGH Conjecture

Prove that if X is the blow-up of the projective plane at n greater than or equal to 10 general points and C is a curve on X, then h^1(O_X(C))=0.

Background

The SHGH Conjecture concerns the vanishing of the first cohomology of line bundles associated with curves on blow-ups of the projective plane at general points. The paper notes that this conjecture implies Nagata’s Conjecture and uses it as motivation for studying bounded cohomology.

Although the paper does not investigate the SHGH Conjecture directly, it states the conjecture explicitly as an unresolved foundational problem related to the bounded cohomology property.

References

Let $C\subseteq X$ be a curve where $X\toP2$ is the blow-up of general points $p_1,\cdots, p_n$ with $n\ge10$. Then $h1(\mathcal O_X(C))=0$.

— Bounded cohomology property on Jacobian elliptic surfaces with simplicial Mori cones  (2609.00592 - Li, 1 Sep 2026) in Introduction, Conjecture 2.5.1

Dumnicki, Küronya, Maclean, and Szemberg show that the Segre--Harbourne--Gimigliano--Hirschowitz conjecture implies the existence of irrational one-point Seshadri constants on suitable blow-ups of the plane.

— Irrational Seshadri constants from dihedral orbits  (2609.26521 - Laface et al., 22 Sep 2026) in Introduction

Nagata's conjecture, originating in his work on Hilbert's fourteenth problem , predicts equality.

— Lifting negative curves from finite fields and Seshadri constants at ten points  (2610.06225 - Pokora et al., 5 Oct 2026) in Section 1, subsection “The Seshadri bound”