Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Could any one explain me how to show that the tangent space at origin of the variety $V=\mathbb{V}(y^2-x^3)$ is equal to full affine plane? They have defined $l$ is a tangent line at $p$ if the multiplicity of $l\cap V$ at $p$ exceeds one. The tangent space $T_p(V)$ of $V$ at $p$ is the union all points lying on the lines tangent to $V$ at $p$

share|cite|improve this question
up vote 2 down vote accepted

In general, if you have a plane curve $V = \mathbb{V}(F(x,y)) \subset \mathbb{A}^2$, and you want to compute the multiplicity of the intersection of $V$ with a line $l$ through $p = (0,0)$, you proceed as follows:

First, you can parameterize your line $l$ as the set of points $\{(at,bt) : t \in \mathbb{K} \}$ for some choice of $(a,b) \in \mathbb{K}^2$, with not both $a$ and $b$ equal to zero. Here I am using $\mathbb{K}$ to denote the field of definition of $V$. (The choice of $(a,b)$ is well-defined only to rescaling -- the lines through $p$ correspond to points of $\mathbb{P}^1$. But you'll see that all the calculations below come out the same if you replace $(a,b)$ with $(\lambda a, \lambda b)$ for any $\lambda \in \mathbb{K}^*$.) The restriction of the polynomial function $F$ to $l$ is given by $F(at,bt)$, which will be an element of $\mathbb{K}[t]$. By definition, the multiplicity of $l \cap V$ at $p = (0,0)$ is the order of vanishing of $t$ in $F(t)$, i.e. the maximal number of powers of $t$ that can be factored out of $F(t)$. Note that the multiplicity if $0$ if and only $p$ does not lie on $V$.

For your particular example, if you follow this procedure, you'll see that every line but one intersects $V$ with multiplicty 2, while one line intersects $V$ with multiplicity $3$. Somethign you should think about: what is that line and how does it relate to the graph of $y^2 = x^3$ in $\mathbb{R}^2$?

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.