Linear systems, base loci, and general members

Fix an algebraically closed field $k$. Let $X$ be a $k$-scheme, let $L$ be an invertible (locally free of rank one) ${\mathcal{O}}_X$-module, and let $W\mathrm{\ne }0$ be a finite-dimensional $k$-linear subspace of $\mathrm{\Gamma }(X,L)$. The subspace $W$ is a linear system on $X$.

Its parameter space is

the set of one-dimensional subspaces of $W$. If ${\dim }_{k}W=r$, a choice of basis identifies $\mathbf{P}(W)$ with ${\mathbf{P}}_k^{r-1}$; give it the projective Zariski topology. A change of basis is an invertible linear coordinate change, which carries homogeneous zero sets to homogeneous zero sets, so this topology does not depend on the chosen basis. In particular, when $r=1$, $\mathbf{P}(W)$ is the one-point space ${\mathbf{P}}_k^0$.

For $0\mathrm{\ne }s\in W$, write $Z(s)\hookrightarrow X$ for its zero subscheme from A section of an invertible sheaf has a canonical zero subscheme. Replacing $s$ by $\lambda s$ for $\lambda \in k^{\times }$ multiplies each local equation by a unit, so it leaves the quotient ideals and the closed subscheme unchanged. Thus $Z(s)$ depends only on the parameter $[s]\in \mathbf{P}(W)$. The base locus of $W$ is the closed subset

It is closed because each $\mathrm{\mid }Z(s)\mathrm{\mid }$ is closed and arbitrary intersections of closed subsets are closed. It is base-point-free when this subset is empty.

A property holds for a general member of $W$ if there is a nonempty Zariski-open subset $U\subseteq \mathbf{P}(W)$ such that every parameter in $U$ has that property.

For a fixed projective embedding $X\subseteq {\mathbf{P}}_k^N$, the hyperplane system is the system cut out by restrictions of degree-one homogeneous forms; the degree-$e$ hypersurface system, for $e\ge 1$, is cut out by restrictions of homogeneous forms of degree $e$. These are viewed as sections of the corresponding powers of the hyperplane line bundle, with forms giving the same section identified.

Vakil, Foundations of Algebraic Geometry Classes 51–52, §3.9 Corollary 3.9, printed p. 10 (PDF page 10, lines 437–441), describes a finite-dimensional base-point-free linear system as a vector space of sections of an invertible sheaf and treats a general section as a point of $\mathbf{P}{H}^0(X,L)$. It asserts that each section gives a closed subscheme, but leaves the Bertini proof to Exercise 3.10; the preceding item supplies the zero-scheme construction. Arapura, Notes on Basic Algebraic Geometry, §5.4, printed pp. 38–39 (PDF pages 38–39, lines 1689–1721), identifies hyperplanes defined by linear forms up to nonzero scalar with the dual projective space and states the smooth hyperplane conclusion on a nonempty open subset. These passages support the projective parameter and “general” conventions and the hyperplane example; the basis-independent topology and arbitrary-subspace wording are made explicit here.