= Parity of real projective plane-curve intersections
{title2=$|Z(f)\cap Z(g)|\equiv ab\pmod2$}
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 $a u$, 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 $a u$. Two transverse zero curves of degrees $a,b$ therefore have intersection parity $\langle ab\,u^2,[\mathbb {RP}^2]_2\rangle=ab$, 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.
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