Moduli space

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. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term 'modulus' is used synonymously with 'parameter'; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces are formal moduli. 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. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term 'modulus' is used synonymously with 'parameter'; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces are formal moduli. Moduli spaces are spaces of solutions of geometric classification problems. That is, the points of a moduli space correspond to solutions of geometric problems. Here different solutions are identified if they are isomorphic (that is, geometrically the same). Moduli spaces can be thought of as giving a universal space of parameters for the problem. For example, consider the problem of finding all circles in the Euclidean plane up to congruence. Any circle can be described uniquely by giving three points, but many different sets of three points give the same circle: the correspondence is many-to-one. However, circles are uniquely parameterized by giving their center and radius: this is two real parameters and one positive real parameter. Since we are only interested in circles 'up to congruence', we identify circles having different centers but the same radius, and so the radius alone suffices to parameterize the set of interest. The moduli space is, therefore, the positive real numbers. Moduli spaces often carry natural geometric and topological structures as well. In the example of circles, for instance, the moduli space is not just an abstract set, but the absolute value of the difference of the radii defines a metric for determining when two circles are 'close'. The geometric structure of moduli spaces locally tells us when two solutions of a geometric classification problem are 'close', but generally moduli spaces also have a complicated global structure as well. For example, consider how to describe the collection of lines in R2 which intersect the origin. We want to assign to each line L of this family a quantity that can uniquely identify it—a modulus. An example of such a quantity is the positive angle θ(L) with 0 ≤ θ < π radians. The set of lines L so parametrized is known as P1(R) and is called the real projective line. We can also describe the collection of lines in R2 that intersect the origin by means of a topological construction. To wit: consider S1 ⊂ R2 and notice that every point s ∈ S1 gives a line L(s) in the collection (which intersects the origin and s). However, this map is two-to-one, so we want to identify s ~ −s to yield P1(R) ≅ S1/~ where the topology on this space is the quotient topology induced by the quotient map S1 → P1(R). Thus, when we consider P1(R) as a moduli space of lines that intersect the origin in R2, we capture the ways in which the members of the family (lines in this case) can modulate by continuously varying 0 ≤ θ < π. The real projective space Pn is a moduli space which parametrizes the space of lines in Rn+1 which pass through the origin. Similarly, complex projective space is the space of all complex lines in Cn+1 passing through the origin. More generally, the Grassmannian G(k, V) of a vector space V over a field F is the moduli space of all k-dimensional linear subspaces of V. The Chow variety Chow(d,P3) is a projective algebraic variety which parametrizes degree d curves in P3. It is constructed as follows. Let C be a curve of degree d in P3, then consider all the lines in P3 that intersect the curve C. This is a degree d divisor DC in G(2, 4), the Grassmannian of lines in P3. When C varies, by associating C to DC, we obtain a parameter space of degree d curves as a subset of the space of degree d divisors of the Grassmannian: Chow(d,P3).