Linear systems of plane curves and their base loci

Work over an algebraically closed field $k$ and fix $d\ge 1$. The space $k[x_0,x_1,x_2{]}_d$ of homogeneous degree-$d$ forms has as a basis the monomials $x_0^i{x}_1^j{x}_2^{d-i-j}$ with $i,j\ge 0$, $i+j\le d$ homogeneous polynomial and homogeneous ideal, Monomials, coefficients, degree in each variable and total degree in $F[x_1,\dots ,x_n]$. The bijection $(i,j)\mapsto \{i,i+j+1\}\subseteq \{0,\dots ,d+1\}$ shows that its dimension is $\left(\genfrac{}{}{0px}{}{d+2}{2}\right)$ The set $[A{]}^k$ of $k$-element subsets and the binomial coefficient $\left(\genfrac{}{}{0px}{}{n}{k}\right):=\mid [n{]}^k\mid$, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis.

A linear system of plane curves of degree $d$ is a linear subspace $W\subseteq k[x_0,x_1,x_2{]}_d$ Linear subspace of a vector space, Vector space over a field. Its elements are the nonzero forms modulo nonzero scalar multiplication, each equipped with its projective zero set:

$[F]⟼V_{+}((F))\subseteq {\mathbf{P}}_k^2,0\mathrm{\ne }F\in W\mathrm{.}$

Here $[F]$ denotes its scalar-equivalence class; scaling does not change its zero set projective space points, projective algebraic set. The dimension of the system is ${\dim }_{k}W-1$. The zero subspace is allowed as the empty system, assigned dimension $-1$ by convention. A pencil has dimension one, equivalently ${\dim }_{k}W=2$.

The base locus is

$\mathrm{Bs}W={\bigcap }_{0\mathrm{\ne }F\in W}V(F)\mathrm{.}$

It is closed as an intersection of projective closed sets projective zariski topology; the empty system has base locus all of ${\mathbf{P}}_k^2$. For a finite set $S\subseteq {\mathbf{P}}_k^2$, the subsystem through $S$ is

$W(-S)=\{F\in W:F(q)=0\text{ for every }q\in S\}\mathrm{.}$

The condition is independent of the chosen nonzero representatives of $q$, since $F(\lambda q)={\lambda }^{d}F(q)$, and is linear in $F$, so $W(-S)$ is a subspace. For the full space $W=k[x_0,x_1,x_2{]}_d$, the dimension is $\left(\genfrac{}{}{0px}{}{d+2}{2}\right)-1$ and the base locus is empty: at any projective point some $x_i\mathrm{\ne }0$, so $x_i^d$ does not vanish there. For $d=1$, a subsystem through three noncollinear points is empty, since the zero set of any nonzero linear form is a line and cannot contain all three.

• This definition retains multiplicities in degree-$d$ equations. A square-free member has reduced degree $d$ in the plane-curve convention degree projective hypersurface. A nonsquarefree form defines the same zero set as its square-free part, whose reduced degree may be smaller. For example $W=⟨x_0^2⟩$ is a degree-two equation system supported on a reduced line of degree one.
• All constructions use explicit finite-dimensional linear algebra and scalar equivalence; no choice principle is used.