Maybe it's worth saying something more about question (2) (like Sándor, I'll stay in the algebraically closed case). This gets very close to the realm of Seshadri constants, see *Positivity in Algebraic Geometry I*, Section 5.

In particular, suppose that we do allow some of the points $x_i$ to be the same. Let $I_{x_i}$ be the ideal defining $x_i$. Then let's define $\mathcal{L}(x_1, \ldots, x_n) = \mathcal{L} \Big/ \Big(\prod_i I_{x_i} \Big)$. In the case that $x_1 = x_2$, a surjectivity $H^0(\mathcal{L}) \to H^0(\mathcal{L}(x_1, \ldots, x_n)) = \mathcal{L}(x_1, \ldots, x_n)$ would in particular mean that the global sections of $\mathcal{L}$ generate $\mathcal{O}_{X,x_1}\big/I_{x_1}^2$ as a vector space (ie, in particular this implies that the global sections separate tangent vectors). If more $x_i$ are equal, then this means that higher order jets (measures of tangency) are separated.

### Sehsadri constants

The notion of Seshadri constants explores the above asymptotically. Indeed, let $Z = \{x_1, \ldots, x_n \}$ denote a collection of closed points (possibly with repeats as above). Define
$$
s(\mathcal{L}^n, Z)
$$
to be the largest integer $k$ such that
$$
H^0(X, \mathcal{L}^n) \to H^0\Big(X, \mathcal{L^n}\big/ \big(\prod_i I_{x_i}^k\big) \Big)
$$
surjects (maybe if we define $kZ$ to be $k$ of the $Z$'s, then this could be compactly written as $H^0(X, \mathcal{L}^n) \to H^0(X, \mathcal{L^n}(kZ))$ using the above notation).

Then we define
$$
\varepsilon(X, \mathcal{L}; Z) = \lim_{k \to \infty} \frac{s(\mathcal{L}^k, Z)}{k}.
$$

The point is, the more positive (ample) $\mathcal{L}$ is at the various points, the bigger the Seshadri constant will be. The question is what this limit

### Applications

Frequently, this is used to actually do the opposite of what you are doing (ie, prove certain ample divisors are in fact very ample, or globally generated). Some common statements include ones like if $X$ is smooth and projective and $\varepsilon(X, \mathcal{L}, x) > \dim X$ for all points $x$ then $\mathcal{L} \otimes \mathcal{O}_X(K_X)$ is globally generated and if $\varepsilon(X, \mathcal{L}, x) > 2 \dim X$ for all $x \in X$ then $\mathcal{L} \otimes \mathcal{O}_X(K_X)$ is very ample.

There are other applications too, like restatements/generalizations of Nagata's conjecture. It's certainly important to also try to understand lower bounds for these constants for ample line bundles: *there aren't any in general, but there are for sufficiently general points*. Anyways, the aforementioned section of Lazarsfeld's book has a nice survey.