Positive square and positive ample intersection force an effective multiple

Given: a field $k$, an integral smooth projective surface $X$ over $k$, an ample invertible sheaf $H$, and an invertible sheaf $L$ with $L\cdot L>0$ and $L\cdot H>0$.

Bilinearity: $\mathbb{Z}$-bilinearity of the intersection product gives $L^{\otimes n}\cdot H=n(L\cdot H)$ and $L^{\otimes n}\cdot L^{\otimes n}=n^2(L\cdot L)$ for every $n\ge 0$; in particular $L^{\otimes n}\cdot H>K_X\cdot H$ for all sufficiently large $n$, with $K_X\cdot H={\omega }_X\cdot H$ (Intersection numbers of Cartier divisors on a smooth projective surface, The surface intersection product is symmetric and bilinear, The canonical divisor of a smooth projective surface, Invertible sheaves, Tensor product of sheaves of modules, Dual of a line bundle is its tensor inverse).

Threshold vanishing: if $M\cdot H>K_X\cdot H$ for an invertible sheaf $M$, then $H^2(X,M)=0$ (Vanishing of top cohomology past the canonical threshold, The canonical divisor of a smooth projective surface).

Riemann-Roch: for every invertible sheaf $M$, $\chi (X,M)=\chi (X,{\mathcal{O}}_X)+\frac{1}{2}(M\cdot M-M\cdot K_X)$ (Riemann-Roch for smooth projective surfaces). The Euler characteristic is the alternating sum $\chi =h^0-h^1+h^2$ of the dimensions $h^q={\dim }_k{H}^q(X,M)$, so $h^0=\chi +h^1-h^2$ with $h^1,h^2\ge 0$ (Euler characteristic of a coherent sheaf).

The Axiom of Choice is inherited from the Riemann-Roch and vanishing suppliers of [F2] and [F3]; the integer $n$ is chosen below from the growth of a quadratic polynomial, no family is selected.