Stable curve

Given an arbitrary scheme ${\displaystyle S}$  and setting ${\displaystyle g\geq 2}$  a stable genus g curve over ${\displaystyle S}$  is defined as a proper flat morphism ${\displaystyle \pi :C\to S}$  such that the geometric fibers are reduced, connected 1-dimensional schemes ${\displaystyle C_{s}}$  such that

1. ${\displaystyle C_{s}}$  has only ordinary double-point singularities
2. Every rational component ${\displaystyle E}$  meets other components at more than ${\displaystyle 2}$  points
3. ${\displaystyle \dim H^{1}({\mathcal {O}}_{C_{s}})=g}$ 

These technical conditions are necessary because (1) reduces the technical complexity (also Picard-Lefschetz theory can be used here), (2) rigidifies the curves so that there are no infinitesimal automorphisms of the moduli stack constructed later on, and (3) guarantees that the arithmetic genus of every fiber is the same. Note that for (1) the types of singularities found in Elliptic surfaces can be completely classified.

One classical example of a family of stable curves is given by the Weierstrass family of curves

${\displaystyle {\begin{matrix}\operatorname {Proj} \left({\frac {\mathbb {Q} [t][x,y,z]}{(y^{2}z-x(x-z)(x-tz)}}\right)\\\downarrow \\\operatorname {Spec} (\mathbb {Q} [t])\end{matrix}}}$ 

where the fibers over every point ${\displaystyle \neq 0,1}$  are smooth and the degenerate points only have one double-point singularity. This example can be generalized to the case of a one-parameter family of smooth hyperelliptic curves degenerating at finitely many points.

In the general case of more than one parameter care has to be taken to remove curves which have worse than double-point singularities. For example, consider the family over ${\displaystyle \mathbb {A} _{s,t}^{2}}$  constructed from the polynomials

${\displaystyle y^{2}=x(x-s)(x-t)(x-1)(x-2)}$ 

since along the diagonal ${\displaystyle s=t}$  there are non-double-point singularities. Another non-example is the family over ${\displaystyle \mathbb {A} _{t}^{1}}$  given by the polynomials

${\displaystyle x^{3}-y^{2}+t}$ 

which are a family of elliptic curves degenerating to a rational curve with a cusp.

One of the most important properties of stable curves is the fact that they are local complete intersections. This implies that standard Serre-duality theory can be used. In particular, it can be shown that for every stable curve ${\displaystyle \omega _{C/S}^{\otimes 3}}$  is a relatively very-ample sheaf; it can be used to embed the curve into ${\displaystyle \mathbb {P} _{S}^{5g-6}}$ . Using the standard Hilbert Scheme theory we can construct a moduli scheme of curves of genus ${\displaystyle g}$  embedded in some projective space. The Hilbert polynomial is given by

${\displaystyle P_{g}(n)=(6n-1)(g-1)}$ 

There is a sublocus of stable curves contained in the Hilbert scheme

${\displaystyle H_{g}\subset {\textbf {Hilb}}_{\mathbb {P} _{\mathbb {Z} }^{5g-6}}^{P_{g}}}$ 

This represents the functor

${\displaystyle {\mathcal {H}}_{g}(S)\cong \left.\left\{{\begin{matrix}&{\text{stable curves }}\pi :C\to S\\&{\text{ with an iso }}\\&\mathbb {P} (\pi _{*}(\omega _{C/S}^{\otimes 3}))\cong \mathbb {P} ^{5g-6}\times S\end{matrix}}\right\}{\Bigg /}{\sim }\right.\cong \operatorname {Hom} (S,H_{g})}$ 

where ${\displaystyle \sim }$  are isomorphisms of stable curves. In order to make this the moduli space of curves without regard to the embedding (which is encoded by the isomorphism of projective spaces) we have to mod out by ${\displaystyle PGL(5g-6)}$ . This gives us the moduli stack

${\displaystyle {\mathcal {M}}_{g}:=[{\underline {H}}_{g}/{\underline {PGL}}(5g-6)]}$ 

• Moduli of algebraic curves
• Stable map of curves

• Artin, M.; Winters, G. (1971-11-01). "Degenerate fibres and stable reduction of curves". Topology. 10 (4): 373–383. doi:10.1016/0040-9383(71)90028-0. ISSN 0040-9383.
• Deligne, Pierre; Mumford, David (1969), "The irreducibility of the space of curves of given genus", Publications Mathématiques de l'IHÉS, 36 (36): 75–109, CiteSeerX 10.1.1.589.288, doi:10.1007/BF02684599, MR 0262240, S2CID 16482150
• Gieseker, D. (1982), Lectures on moduli of curves (PDF), Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 69, Published for the Tata Institute of Fundamental Research, Bombay, ISBN 978-3-540-11953-1, MR 0691308
• Harris, Joe; Morrison, Ian (1998), Moduli of curves, Graduate Texts in Mathematics, vol. 187, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98429-2, MR 1631825