Cohomological Amplitude of Moduli Spaces of Curves

A trial analysis of the following topic from Conference on [Algebraic Groups] and [Representation Theory].

Topic: Cohomological Amplitude of Moduli Spaces of Curves

For a scheme one has [a notion of [cohomological excess]]. The speaker first defines and discusses this notion and then he states the main result which says that the [universal curve of genus] g > 0 has cohomological excess at most g-1. The speaker shows that this is an [algebro-geometric] strengthening of [Harer’s theorems] on the [[virtual cohomological dimension] of the [mapping class group]] and implies a [theorem of Diaz] on the [maximal dimension] of a [complete subvariety] of [a moduli space of curves]. He also discusses some ingredients of the proof.



Step 1: Determine the main theme.

Main Theme: Mathematics

Step 2: Find out the meaning of unknown terms. (What is it?)

T1.1 An algebraic group is a [group] that is an [algebraic variety], such that the multiplication and inverse are given by [regular functions] on the variety.

T1.2 Representation theory is a branch of mathematics that studies [abstract algebraic structures] by representing their elements as linear transformations of [vector spaces], and studies modules over these [abstract algebraic structures].

T1.3 The cohomological excess of a nonempty [algebraic variety] X, denoted ce(X), is the maximum of the integers ccd(W)−dimW, where W runs over all the [Zariski closed subsets] W ⊂ X.

T1.4 Genus: A [topologically invariant property] of a surface defined as the largest number of non-intersecting simple closed curves that can be drawn on the surface without separating it. Roughly speaking, it is the number of holes in a surface. The genus of a surface, also called the geometric genus, is related to the Euler characteristic X.

T1.5 Algebraic geometry is a branch of mathematics, classically studying properties of the sets of zeros of polynomial equations. Modern algebraic geometry is based on more abstract techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.

T1.6 Harer’s theorems (I am not quite sure about it.)

T1.7 In abstract algebra, cohomological dimension is an invariant which measures the homological complexity of representations of a group. It has important applications in geometric group theory, topology, and algebraic number theory.

T1.8 In the sub-field of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.

T1.9 theorem of Diaz  (I am not quite sure about it.)

T1.10 In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. A moduli space of (algebraic) curves is a geometric space whose points represent isomorphism classes of algebraic curves.

T1.11 complete subvariety => An algebraic variety is an irreducible algebraic set, i.e. one which is not the union of two other algebraic sets which are the sets of solutions of a system of polynomial equations.

Step 3: Find out the unknown common terms or categories in the first round of elaboration.

Directly Connected to the Topic

1. Group
2. Algebraic structure
3. Geometric space
4. Algebraic curve
5. Cohomology

Indirectly Connected to the Topic

1. Algebraic set => Algebraic variety
2. Algebraic geometry
3. Topological space 
4. Geometric topology
5. Vector space

Step 4: Elaborate the unknown common terms. (What is it?)

T2.1 A group is an algebraic structure consisting of a set together with a binary operation.

T2.2 An algebraic structure generally refers to an arbitrary set with one or more finitary operations defined on it.

T2.3 Geometric space is a special topological space. Topological space is a mathematical structure that allows the formal definition of concepts such as convergence, connectedness, and continuity. Examples of mathematical structure are order, measure, metric, geometry, topology.

T2.4 Algebraic curves are the simplest objects of Euclidean geometry (T2.3) that cannot be defined by linear properties.


T2.5 Cohomology is a general term for a sequence of abelian groups (T2.1) defined from a co-chain complex. Chain complex is an algebraic mean of representing the relationships between the cycles and boundaries in various dimensions of a topological space. (T2.3)

T2.6 Algebraic geometry is a branch of mathematics, classically studying properties of the sets of zeros of polynomial equations.

T2.7 Geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.

T2.8 A vector space is a mathematical structure (T2.3) formed by a collection of elements called vectors, which may be added together and multiplied by numbers, called scalars in this context.


T3.1 A manifold is a second countable Hausdorff space that is locally homeomorphic to Euclidean space. e.g. curve and line (1-manifold),  sphere and torus (2-manifold).


Step 5: Try to reduce the key statement.
The cohomological excess (of nonempty irreducible algebraic set)
is a theorem on the dimension of the discrete sets (with binary operation) of 'symmetries' of the topological space and
implies another theorem on the maximal dimension of a irreducible algebraic set of a geometric space whose points represent isomorphism classes of algebraic curves.

Step 6: Simplify the key statement.
A theory on the dimension of the discrete sets with binary operation in topological space
implies
another theorem on the maximal dimension of a irreducible algebraic set of a geometric space.


Can we understand it?