Zariski tangent space

Imagine you have a point on a curve drawn on a piece of paper. If you were to zoom in really closely on that point, the curve would look like a straight line. This straight line is called the tangent line and it tells you the direction the curve is going at that point.

Similarly, in algebraic geometry, if you have a point on an algebraic variety (like a curve), you can look at the set of functions (polynomials) that vanish at that point. This set is called the ideal of the point and it tells you information about the point and its neighboring points on the variety.

The Zariski tangent space is a way to describe the directions in which the variety (curve) is going at the point. It is the set of all derivatives (tangent vectors) of functions in the ideal that vanish at the point. These derivatives are like arrows that point in different directions away from the point.

So, just like the tangent line on a curve tells you the direction the curve is going, the Zariski tangent space tells you the possible directions the variety is going at the point. By studying the Zariski tangent space, we can learn more about the variety and how it behaves near the point.