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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 $p$

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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 $\mathbb{R}^2$?

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