Say we have an affine variety $V \subset \mathbb{C}^n$, and suppose we intersect $V$ with a hyperplane $H$, possibly not in general position. Is it possible for the degree of $V \cap H$ to be larger than the degree of $V$?



Can the degree of an affine variety increase after intersecting with a hyperplane?
Yes, it is possible for the degree to increase. Say $V \subset \mathbb{C}^3$ is reducible: a union of a curve of degree $d$ plus one more point that doesn't lie on the curve. Then $V$ has degree $d$. But if $H$ is any plane through that extra point, then the intersection of $V$ with $H$ has degree $d+1$: the one point, plus $d$ points of intersection with the curve.


More generally, degree only sees the highest-dimensional component of $V$, but intersection with a hyperplane can bring up lower-dimensional components, raising the degree.


