= Genus one tangent incidence curve of two conics
{title2=$E=\{(\ell,x):\ell\text{ tangent to }C_1,\ x\in C_2\cap\ell\}$}
If two <smooth plane conics> over $\mathbb C$ meet at four distinct points, their tangent incidence curve is a smooth projective <genus one curve>. Projection to $C_2$ is a double cover branched at those four intersections. The tangent equation becomes a quadratic whose discriminant cuts out $C_1\cap C_2$; its four simple zeros make the cover smooth and connected. The <Riemann-Hurwitz formula> then gives genus one.
Back to article page
