Zariski tangent space

Summary

The Zariski tangent space of a variety at a point is the linear space of directions in which the defining equations vanish to first order. For a variety in $k^n$ it is the kernel of the Jacobian matrix of the defining polynomials at the point. Intrinsically it is the dual of $\mathfrak m / \mathfrak m^2$, where $\mathfrak m$ is the maximal ideal of the point. Its dimension is at least the dimension of the variety, with equality exactly at the nonsingular points.