The Alexander–Hirschowitz property for ample line bundles on smooth projective varieties

Let $X$ be a smooth projective variety and let $\cal{L}$ be an ample line bundle. Say that $(X,\mathcal{L})$ satisfies the Alexander–Hirschowitz property (AH) if, for every positive integer $k$, a general collection of double points $p_1,\ldots,p_k\in X$ imposes independent conditions on $|\mathcal{L}|$; equivalently, the linear system of divisors singular at all these points has dimension

$$\max\left(0,\chi(X,\mathcal{L}^{\otimes d})-k(\dim(X)+1)\right)-1.$$

Alexander–Hirschowitz conjecture. Suppose that $X$ is a smooth projective variety of any dimension and $\mathcal{L}$ is ample. Then $(X,\mathcal{L})$ satisfies AH.

The paper proves this property for smooth projective surfaces and conjectures the analogous statement in arbitrary dimension. The source notes that a result subsuming both the main theorem and this conjecture had already been proved, so the conjecture is solved.