Zariski tangent space

Remarks

If $I(X) = (f_1, \dots, f_r)$, then $T_p X$ is the kernel of the Jacobian matrix $\left(\frac{\partial f_j}{\partial x_i}(p)\right)$, because $d_p(gf) = g(p) \, d_p f$ when $f(p) = 0$.

Intrinsic form: let $\mathfrak m_p$ be the maximal ideal of the coordinate ring $k[X]$ of functions vanishing at $p$. Then $T_p X$ is isomorphic to the dual vector space of $\mathfrak m_p / \mathfrak m_p^2$. The same space is the dual of the cotangent space of the local ring $k[X]_{\mathfrak m_p}$, so $T_p X$ only depends on a neighbourhood of $p$ and is defined for every variety.

Example: for the parabola $X = V(y - x^2)$ at $p = (a, a^2)$, $d_p(y - x^2) = -2a \, v_1 + v_2$, so $T_p X$ is the line $v_2 = 2a v_1$, of dimension $1$.

Example: for the cusp $X = V(y^2 - x^3)$ at the origin, the polynomial $y^2 - x^3$ generates $I(X)$ and both of its partial derivatives vanish at $(0, 0)$, so $T_{(0,0)} X = k^2$ has dimension $2 > \dim X = 1$.

One always has $\dim T_p X \ge \dim X$.