Moduli Space

Good moduli spaces are finite type over the base theorem 4 16 xi and the proof that good moduli spaces are unique in the category of algebraic spaces theorem 6 6.

Moduli space. We give a number of examples of moduli stacks in section 8 admitting good moduli spaces including the moduli of semi stable sheaves and alternative com pactifications of mg. Moduli space can be thought of. Moduli spaces in topology are often referred to as classifying spaces.

The actual definition of a fine moduli space is stronger than saying that points classify something and requires families of points on the moduli space to correspond to families of the kind of object you re trying to classify in a very precise way. Moduli space in algebraic geometryclassification problems an algebraic variety or other appropriate space in other parts of geometry whose points correspond to the equivalence classes of the objects to be classified in some natural way. In mathematics in particular algebraic geometry a moduli space is a geometric space usually a scheme or an algebraic stack whose points represent algebro geometric objects of some fixed kind or isomorphism classes of such objects.

This is an example of a general phenomenon. The rough idea is but see the caveat below that the term moduli space is essentially a synonym for representing object and for classifying space. Since the moduli space is a point these must induce the same map to the moduli space despite being non isomorphic.

People tend to say classifying space when in the context of topology and they tend to say moduli space when in a context of complex geometry or algebraic geometry. The corresponding coarse moduli space of c g is m g 1 so the coarse moduli space of curves with one marked point. In algebraic geometry a moduli space is a geometric space usually a scheme or an algebraic stack whose points represent algebro geometric objects of some fixed kind or isomorphism classes of such objects.