= Pencil of conics through four points
{title2=$\{\lambda B_0+\mu B_1:[\lambda:\mu]\in\mathbb P^1\}$}
Four <projective points> in <general linear position> in the complex projective plane impose four independent linear conditions on the six coefficients of a <quadratic polynomial>. To see independence, put them at the three coordinate points and $[1:1:1]$ using a <projective frame>. The remaining equations are $Axy+Bxz+Cyz=0$ with $A+B+C=0$, a two-dimensional <vector space>. Projectivizing its nonzero elements gives a pencil. It has nonsingular bases, for example the forms with coefficient triples $(1,1,-2)$ and $(1,-2,1)$, because the determinant of the symmetric coefficient matrix is $ABC/4$. Its three singular members are exactly the three pairs of lines obtained by partitioning the four base points into two pairs.
Back to article page
