# Can the degree of an affine variety increase after intersecting with a hyperplane?

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 $$V$$?

• How do you define the degree of $V$ of $V$ is not a hypersurface? – YCor Feb 13 at 23:32
• The definition I'm using is that if $V$ has dimension $d$, then the degree of $V$ is equal to the number of intersection point of $V$ with $d$ hyperplanes in general position. (Including intersections at infinity and intersection multiplicity). – Powerspawn Feb 13 at 23:36
• you mean $n-d$ hyperplanes – Abdelmalek Abdesselam Feb 13 at 23:43
• @AbdelmalekAbdesselam The hyperplanes have codimension $1$, so each intersection reduces the dimension of $V$ by $1$, so we need to intersect with $d$ hyperplanes total. – Powerspawn Feb 13 at 23:49
• oops you're right got mixed up between what is dimension and what is codimension – Abdelmalek Abdesselam Feb 13 at 23:55

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.