8.1 Moduli spaces of curves and stable curves

Moduli spaces of curves are geometric spaces that classify different types of curves. They're crucial in algebraic geometry, helping us understand how curves behave in families and how they can degenerate into singular curves.

Stable curves are a key concept in compactifying moduli spaces. By including these mildly singular curves, we can create a complete space that captures all possible limits of smooth curves, giving us a more comprehensive view of curve behavior.

Moduli spaces of curves

Definition and notation

• Moduli spaces are geometric spaces that parametrize equivalence classes of geometric objects (algebraic curves, vector bundles)
• The moduli space of curves of genus $g$, denoted $M_g$, is the space parametrizing isomorphism classes of smooth, projective curves of genus $g$
• $M_g$ is not compact, as families of smooth curves can degenerate to singular curves not represented in $M_g$

Stable curves and compactification

• Stable curves are singular curves with mild singularities (only nodes) that arise as limits of smooth curves in families
  • A stable curve has only nodes as singularities and has finite automorphism group
• The moduli space of stable curves of genus $g$, denoted $\bar{M}_g$, is a compactification of $M_g$ that includes stable curves as boundary points
• Adding stable curves to compactify $M_g$ reflects the properness of the moduli problem: limits of families of smooth curves should be represented in the compactification

Constructing moduli spaces

Dimension and construction methods

• The dimension of the moduli space $M_g$ is:
  • $3g-3$ for $g \geq 2$
  • $1$ for $g = 1$
  • $0$ for $g = 0$
• $M_g$ can be constructed as a quotient of a Hilbert scheme or Chow variety parametrizing curves in projective space by the action of the projective linear group
• The Deligne-Mumford compactification $\bar{M}_g$ is constructed by adding stable curves as limits of smooth curves
• Teichmüller theory provides an alternative construction of $M_g$ as a quotient of Teichmüller space by the action of the mapping class group

Stack structure

• $M_g$ has a natural stack structure, reflecting the presence of curves with automorphisms
• The stack structure encodes the automorphism groups of the curves parametrized by the moduli space
• Considering $M_g$ as a stack leads to a richer geometric structure and allows for a finer classification of curves

Geometry of moduli spaces

Geometric properties

• $M_g$ is an irreducible, quasi-projective variety of dimension $3g-3$ for $g \geq 2$
• The compactification $\bar{M}_g$ is a projective variety with normal crossings boundary divisor $\partial \bar{M}_g = \bar{M}_g - M_g$ parametrizing stable curves
• The boundary $\partial \bar{M}_g$ stratifies into components $\Delta_i$ parametrizing stable curves with nodes of given topological type

Vector bundles and intersection theory

• $M_g$ and $\bar{M}_g$ carry natural vector bundles:
  • The Hodge bundle, whose fibers are the spaces of holomorphic differentials on the curves
  • Tautological line bundles, which arise from the universal curve over the moduli space
• Moduli spaces of curves have rich intersection theory, with tautological classes satisfying Witten-Kontsevich recursions
• Intersection numbers on $\bar{M}_g$ encode information about the enumerative geometry of curves (counting curves satisfying incidence conditions)

Relation to mapping class groups

• $M_g$ and $\bar{M}_g$ are classifying spaces for:
  • Mapping class groups (groups of isotopy classes of diffeomorphisms) of surfaces of genus $g$
  • Outer automorphism groups of free groups on $g$ generators
• The cohomology of $M_g$ and $\bar{M}_g$ is closely related to the cohomology of mapping class groups and outer automorphism groups of free groups

Stable curves and compactifications

Definition and classification

• A stable curve is a connected, projective curve with nodes as singularities and finite automorphism group
• Stable curves can be classified by:
  • Genus: the arithmetic genus, which is the genus of the normalization plus the number of nodes
  • Number of nodes: the number of singular points on the curve
  • Genera of irreducible components: the geometric genera of the smooth components obtained by normalizing the nodes

Limits of smooth curves

• Families of smooth curves acquire stable curves as limits under specialization
  • A one-parameter family of smooth curves of genus $g > 1$ has a stable limit after a finite base change (stable reduction theorem)
  • The stable reduction theorem states that any family of curves admits a stable limit after base change and birational modifications
• The boundary $\partial \bar{M}_g$ parametrizes stable curves and has a stratification by topological type of the curves
  • Each stratum corresponds to a dual graph encoding the genera of the components and the configuration of nodes
  • The codimension of a stratum is equal to the number of nodes of the corresponding stable curves