# Why we study Geometric invariant theory?

I am trying to learn Geometric invariant theory like it was introduced by Mumford. But I do not have a strong motivation and so I want to know the reason of studying Geometric invariant theory. I just know Geometric invariant theory plays an important role in the construction of moduli spaces.

What's a reason for studying Geometric invariant theory? And, what are related (big or small) problems with Geometric invariant theory in algebraic geometry?

• One reason for studying GIT would be the construction of quotients (and hence of various types of moduli spaces). Mumford showed that if $X \subseteq \Bbb{P}(V)$ is a projective variety with $V$ a linear representation of a linearly reductive group $G$, such that $G$ acts on $X$ via restriction, then a good quotient of the semistable locus $X^{ss}$ exists. This idea was used by Mumford, Gieseker, etc to prove that a coarse moduli space $M_g$ exists. Nowadays, there are more fancy methods to prove this coarse moduli space exists, such as the Keel-Mori Theorem. – Ben Lim May 3 '15 at 16:59
Geometric invariant theory also classifies the relations between the invariants (and relations between the relations etc). For example, the ideal of relations between the invariants of two vectors in euclidean geometry is generated by the relation $$\parallel u \parallel^2 \parallel v \parallel^2 - \parallel u \times v\parallel^2 - \parallel u | v \ \parallel^2$$ It can be shown that any theorem of projective plane geometry (either true or false) is equivalent to a relation between the invariants of the geometry. Here, the term "Theorem" should be understood in a geometric sense, as a (finite) succession of projective transformations that build points and lines from a given (finite) set of initial conditions.
For example the projective Pappus theorem is equivalent to the following (true) relation $$[ (b \wedge c') \wedge (b'\wedge c), (c \wedge a')\wedge (c' \wedge a), (a \wedge b')\wedge (a'\wedge b)] = [b',c',a][c',a',b][a',b',c][a,b,c] - [b,c,a'][c,a,b'][a,b,c'][a',b',c']$$ where $a,b,c,a',b',c'$ are ( projective coordinates of the) points in the plane, $a\wedge b$ is (the coordinates in the dual of) the line determined by the points $a,b$, the wedge of two lines is the intersection point of the lines, and the bracket is the determinant of the three vectors representing the points in the projective plane. This may appear as a pretty complex relation between the coordinates of the points when expanded in polynomial form, but geometric invariant theory provides a neat classification of these relations.