 General position for projective lines

ID: general-position-for-projective-lines

Distinct lines in the projective plane are in general position when no three are concurrent. Their union is a reduced plane curve with exactly

(2 d)

singular points, all ordinary pairwise crossings, if there are

d

lines. An explicit family is

y = t_{i} x + t_{i}^{2} z

for distinct complex

t_{i}

: the intersection of the

i

th and

j

th lines is

[- t_{i} - t_{j} : - t_{i} t_{j} : 1]

, and substituting it into a third equation gives

- (t_{k} - t_{i}) (t_{k} - t_{j}) \neq = 0

 New to topics? Read the docs here!