# Zariski tangent space

In algebraic geometry, the **Zariski tangent space** is a construction that defines a tangent space at a point *P* on an algebraic variety *V* (and more generally). It does not use differential calculus, being based directly on abstract algebra, and in the most concrete cases just the theory of a system of linear equations.

For example, suppose given a plane curve *C* defined by a polynomial equation

and take *P* to be the origin (0,0). Erasing terms of higher order than 1 would produce a 'linearised' equation reading

We have two cases: *L* may be 0, or it may be the equation of a line. In the first case the (Zariski) tangent space to *C* at (0,0) is the whole plane, considered as a two-dimensional affine space. In the second case, the tangent space is that line, considered as affine space. (The question of the origin comes up, when we take *P* as a general point on *C*; it is better to say 'affine space' and then note that *P* is a natural origin, rather than insist directly that it is a vector space.)

It is easy to see that over the real field we can obtain *L* in terms of the first partial derivatives of *F*. When those both are 0 at *P*, we have a singular point (double point, cusp or something more complicated). The general definition is that *singular points* of *C* are the cases when the tangent space has dimension 2.

(One often defines the tangent and cotangent spaces for a manifold in the analogous manner.)

If *V* is a subvariety of an *n*-dimensional vector space, defined by an ideal *I*, then *R = F _{n}* /

*I*, where

*F*is the ring of smooth/analytic/holomorphic functions on this vector space. The Zariski tangent space at

_{n}*x*is

where *m _{n}* is the maximal ideal consisting of those functions in

*F*vanishing at

_{n}*x*.

If *R* is a Noetherian local ring, the dimension of the tangent space is at least the dimension of *R*:

*R* is called regular if equality holds. In a more geometric parlance, when *R* is the local ring of a variety *V* at a point *v*, one also says that *v* is a regular point. Otherwise it is called a **singular point**.

The tangent space has an interpretation in terms of *K*[*t*]*/*(*t ^{2}*), the dual numbers for

*K*; in the parlance of schemes, morphisms from

*Spec*

*K*[

*t*]

*/*(

*t*) to a scheme

^{2}*X*over

*K*correspond to a choice of a rational point

*x ∈ X(k)*and an element of the tangent space at

*x*.

^{[3]}Therefore, one also talks about

**tangent vectors**. See also: tangent space to a functor.

Zariski, Oscar (1947). . *Trans. Amer. Math. Soc*. **62**: 1–52. doi:. MR . Zbl .