Parity of real projective plane-curve intersections ($|Z(f)\cap Z(g)|\equiv ab(\mathrm{mod}2)$)

A degree-$a$ homogeneous real polynomial defines a section of the $a$th tensor power of the dual real tautological line bundle over ${\mathbb{RP}}^2$. Its first Stiefel–Whitney class is $au$, where $u$ generates $H^1({\mathbb{RP}}^2;{\mathbb{F}}_2)$. By parity of zeros of a real line-bundle section, a generic zero curve has mod-two dual class $au$. Two transverse zero curves of degrees $a,b$ therefore have intersection parity $⟨ab{u}^2,[{\mathbb{RP}}^2{]}_2⟩=ab$, since two distinct projective lines intersect once. For example, the line $z=0$ and the conic $x^2+y^2-z^2=0$ are disjoint in the real projective plane, although their degree product is two.