Intersection multiplicity with a non-tangent line

Question

Let $V$ be an algebraic variety and $x$ a point of $V$ and $\ell$ a line passing through $x$ such that $\ell$ is not included in the tangeant space $T_{V,x}$. It seems intuitive that the intersection multiplicity $i(\ell,V;x)$ should be $1$, but how to prove it?

Answer 1

Your variety $V$ should be embedded in projective (or affine) space, since otherwise you cannot speak of a "line". As your question is local, you can assume that $V$ is a subvariety of the affine space $\mathbb{A}^n$ and your point $x$ is the origin $0$. Clearly the tangent space $T_0 \ell$ of $\ell$ in $0$ is just the line $\ell$ itself, when we consider $T_0 \ell$ as a subspace of $T_0 \mathbb{A}^n = \mathbb{A}^n$. If I understand your assumption correctly, the intersection of $T_0 \ell$ and $T_0 V$ inside $T_0 \mathbb{A}^n$ is just $0$. Thus after blowing up $\mathbb{A}^n$ in $0$, the strict transform of $V$ and $\ell$ do not intersect in the exceptional divisor. Hence, $i(\ell, V; x) = 1$.