@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.

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).