More open substacks of Curves

Decompose into a piece for each genus....

moduli-curves.tex

@@ -666,6 +666,110 @@ \section{Curves with finite reduced automorphism groups}
 
 
 
+\section{Cohen-Macaulay curves}
+\label{section-CM}
+
+\noindent
+There is an open substack of $\Curvesstack$ parametrizing
+the Cohen-Macaulay ``curves''.
+
+\begin{lemma}
+\label{lemma-CM-curves}
+There exist an open substack $\Curvesstack^{CM} \subset \Curvesstack$
+such that
+\begin{enumerate}
+\item given a family of curves $X \to S$ the following are equivalent
+\begin{enumerate}
+\item the classifying morphism $S \to \Curvesstack$ factors
+through $\Curvesstack^{CM}$,
+\item the morphism $X \to S$ is Cohen-Macaulay,
+\end{enumerate}
+\item given a scheme $X$ proper over a field $k$ with $\dim(X) \leq 1$
+the following are equivalent
+\begin{enumerate}
+\item the classifying morphism $\Spec(k) \to \Curvesstack$ factors
+through $\Curvesstack^{CM}$,
+\item $X$ is Cohen-Macaulay.
+\end{enumerate}
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+Let $f : X \to S$ be a family of curves. By
+More on Morphisms of Spaces, Lemma
+\ref{spaces-more-morphisms-lemma-flat-finite-presentation-CM-open}
+the set
+$$
+W = \{x \in |X| : f \text{ is Cohen-Macaulay at }x\}
+$$
+is open in $|X|$ and formation of this open commutes with arbitrary
+base change. Since $f$ is proper the subset
+$$
+S' = S \setminus f(|X| \setminus W)
+$$
+of $S$ is open and $X \times_S S' \to S'$ is Cohen-Macaulay.
+Moreover, formation of $S'$ commutes with arbitrary base
+change because this is true for $W$
+Thus we get the open substack with the desired properties
+by the method discussed in Section \ref{section-open}.
+\end{proof}
+
+\begin{lemma}
+\label{lemma-CM-1-curves}
+There exist an open substack $\Curvesstack^{CM, 1} \subset \Curvesstack$
+such that
+\begin{enumerate}
+\item given a family of curves $X \to S$ the following are equivalent
+\begin{enumerate}
+\item the classifying morphism $S \to \Curvesstack$ factors
+through $\Curvesstack^{CM, 1}$,
+\item the morphism $X \to S$ is Cohen-Macaulay and has
+relative dimension $1$ (Morphisms of Spaces, Definition
+\ref{spaces-morphisms-definition-relative-dimension}),
+\end{enumerate}
+\item given a scheme $X$ proper over a field $k$ with $\dim(X) \leq 1$
+the following are equivalent
+\begin{enumerate}
+\item the classifying morphism $\Spec(k) \to \Curvesstack$ factors
+through $\Curvesstack^{CM, 1}$,
+\item $X$ is Cohen-Macaulay and every irreducible component of $X$
+has dimension $1$.
+\end{enumerate}
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+By Lemma \ref{lemma-CM-curves} it is clear that we have
+$\Curvesstack^{CM, 1} \subset \Curvesstack^{CM}$
+if it exists. Let $f : X \to S$ be a family of curves
+such that $f$ is a Cohen-Macaulay morphism. By
+More on Morphisms of Spaces, Lemma
+\ref{spaces-more-morphisms-lemma-lfp-CM-relative-dimension}
+we have a decomposition
+$$
+X = X_0 \amalg X_1
+$$
+by open and closed subspaces such that $X_0 \to S$ has relative
+dimension $0$ and $X_1 \to S$ has relative dimension $1$.
+Since $f$ is proper the subset
+$$
+S' = S \setminus f(|X_0|)
+$$
+of $S$ is open and $X \times_S S' \to S'$ is Cohen-Macaulay
+and has relative dimension $1$.
+Moreover, formation of $S'$ commutes with arbitrary base
+change because this is true for the decomposition above
+(as relative dimension behaves well with respect to base
+change, see Morphisms of Spaces, Lemma
+\ref{spaces-morphisms-lemma-dimension-fibre-after-base-change}).
+Thus we get the open substack with the desired properties
+by the method discussed in Section \ref{section-open}.
+\end{proof}
+
+
+
+
+
 
 \section{Geometrically reduced curves}
 \label{section-geometrically-reduced}
@@ -714,58 +818,174 @@ \section{Geometrically reduced curves}
 by the method discussed in Section \ref{section-open}.
 \end{proof}
 
+\begin{lemma}
+\label{lemma-geomred-in-CM}
+We have $\Curvesstack^{geomred} \subset \Curvesstack^{CM}$
+as open substacks of $\Curvesstack$.
+\end{lemma}
 
+\begin{proof}
+This is true because a reduced Noetherian scheme of
+dimension $\leq 1$ is Cohen-Macaulay. See
+Algebra, Lemma \ref{algebra-lemma-criterion-reduced}.
+\end{proof}
 
 
 
-\section{Cohen-Macaulay curves}
-\label{section-CM}
+
+
+
+\section{Geometrically reduced and connected curves}
+\label{section-geometrically-reduced-connected}
 
 \noindent
 There is an open substack of $\Curvesstack$ parametrizing
-the Cohen-Macaulay ``curves''.
+the geometrically reduced and connected ``curves''.
+We will get rid of $0$-dimensional objects right away.
 
 \begin{lemma}
-\label{lemma-CM-curves}
-There exist an open substack $\Curvesstack^{CM} \subset \Curvesstack$
+\label{lemma-geometrically-reduced-connected-1-curves}
+There exist an open substack $\Curvesstack^{grc, 1} \subset \Curvesstack$
 such that
 \begin{enumerate}
 \item given a family of curves $X \to S$ the following are equivalent
 \begin{enumerate}
 \item the classifying morphism $S \to \Curvesstack$ factors
-through $\Curvesstack^{CM}$,
-\item the morphism $X \to S$ is Cohen-Macaulay,
+through $\Curvesstack^{grc, 1}$,
+\item the geometric fibres of the morphism $X \to S$ are
+reduced, connected, and have dimension $1$,
 \end{enumerate}
 \item given a scheme $X$ proper over a field $k$ with $\dim(X) \leq 1$
 the following are equivalent
 \begin{enumerate}
 \item the classifying morphism $\Spec(k) \to \Curvesstack$ factors
-through $\Curvesstack^{CM}$,
-\item $X$ is Cohen-Macaulay.
+through $\Curvesstack^{grc, 1}$,
+\item $X$ is geometrically reduced, geometrically connected,
+and has dimension $1$.
 \end{enumerate}
 \end{enumerate}
 \end{lemma}
 
 \begin{proof}
-Let $f : X \to S$ be a family of curves. By
-More on Morphisms of Spaces, Lemma
-\ref{spaces-more-morphisms-lemma-flat-finite-presentation-CM-open}
-the set
+By Lemmas \ref{lemma-geometrically-reduced-curves},
+\ref{lemma-geomred-in-CM}, \ref{lemma-CM-curves}, and \ref{lemma-CM-1-curves}
+it is clear that we have
 $$
-W = \{x \in |X| : f \text{ is Cohen-Macaulay at }x\}
+\Curvesstack^{geomredcon}
+\subset
+\Curvesstack^{geomred} \cap \Curvesstack^{CM, 1}
 $$
-is open in $|X|$ and formation of this open commutes with arbitrary
-base change. Since $f$ is proper the subset
+if it exists. Let $f : X \to S$ be a family of curves such that $f$ is
+Cohen-Macaulay, has geometrically reduced fibres, and
+has relative dimension $1$. By
+More on Morphisms of Spaces, Lemma
+\ref{spaces-more-morphisms-lemma-stein-factorization-etale}
+in the Stein factorization
 $$
-S' = S \setminus f(|X| \setminus W)
+X \to T \to S
 $$
-of $S$ is open and $X \times_S S' \to S'$ is Cohen-Macaulay.
-Moreover, formation of $S'$ commutes with arbitrary base
-change because this is true for $W$
+the morphism $T \to S$ is \'etale. This implies that
+there is an open and closed subscheme $S' \subset S$
+such that $X \times_S S' \to S'$ has geometrically
+connected fibres (in the decomposition of
+Morphisms, Lemma \ref{morphisms-lemma-finite-locally-free}
+for the finite locally free morphism $T \to S$
+this corresponds to $S_1$).
+Formation of this open commutes with arbitrary base change
+because the number of connected components of geometric
+fibres is invariant under base change (it is also true
+that the Stein factorization commutes with base change
+in our particular case but we don't need this to conclude).
 Thus we get the open substack with the desired properties
 by the method discussed in Section \ref{section-open}.
 \end{proof}
 
+\begin{lemma}
+\label{lemma-genus}
+Let $f : X \to S$ be a family of curves whose geometric fibres are
+reduced, connected, and have dimension $1$. Then
+\begin{enumerate}
+\item $f_*\mathcal{O}_X = \mathcal{O}_S$ and this holds universally,
+\item $R^1f_*\mathcal{O}_X$ is a finite locally free $\mathcal{O}_S$-module,
+\item for any morphism $h : S' \to S$ if $f' : X' \to S'$ is the base change,
+then $h^*(R^1f_*\mathcal{O}_X) = R^1f'_*\mathcal{O}_{X'}$.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+Part (1) holds by Derived Categories of Spaces,
+Lemma \ref{spaces-perfect-lemma-proper-flat-geom-red-connected}.
+By cohomology and base change (more precisely by
+Derived Categories of Spaces, Lemma
+\ref{spaces-perfect-lemma-flat-proper-perfect-direct-image-general})
+we see that $E = Rf_*\mathcal{O}_X$ is a perfect object of the
+derived category of $S$ and that its formation commutes with
+arbitrary change of base.
+By part (1) we can locally on $S$ write
+$E = \mathcal{O}_S \oplus E'$ in $D(\mathcal{O}_S)$
+with $E' = \tau_{\geq 1}E$ a perfect object
+with tor amplitude in $[1, \infty)$, see
+More on Algebra, Lemma \ref{more-algebra-lemma-better-cut-complex-in-two}.
+For $s \in S$ we have
+$$
+H^i(E' \otimes_{\mathcal{O}_S}^\mathbf{L} \kappa(s)) =
+H^i(X_s, \mathcal{O}_{X_s})
+\text{ for }i \geq 1
+$$
+since formation of $E$ commutes with arbitrary base change.
+This is zero unless $i = 1$ since $X_s$ is a $1$-dimensional
+Noetherian scheme, see
+Cohomology, Proposition \ref{cohomology-proposition-vanishing-Noetherian}.
+Then $E' = H^1(E')[-1]$ and $H^1(E')$ is finite locally free
+by More on Algebra, Lemma
+\ref{more-algebra-lemma-lift-perfect-from-residue-field}.
+Since everything is compatible with base change we
+also see that (3) holds.
+\end{proof}
+
+\begin{lemma}
+\label{lemma-one-piece-per-genus}
+There is a decomposition into open and closed substacks
+$$
+\Curvesstack^{grc, 1} = \coprod\nolimits_{g \geq 0} \Curvesstack^{grc, 1}_g
+$$
+where each $\Curvesstack^{grc, 1}_g$ is characterized as follows:
+\begin{enumerate}
+\item given a family of curves $f : X \to S$ the following are equivalent
+\begin{enumerate}
+\item the classifying morphism $S \to \Curvesstack$ factors
+through $\Curvesstack^{grc, 1}_g$,
+\item the geometric fibres of the morphism $f : X \to S$ are
+reduced, connected, of dimension $1$ and
+$R^1f_*\mathcal{O}_X$ is a locally free $\mathcal{O}_S$-module
+of rank $g$,
+\end{enumerate}
+\item given a scheme $X$ proper over a field $k$ with $\dim(X) \leq 1$
+the following are equivalent
+\begin{enumerate}
+\item the classifying morphism $\Spec(k) \to \Curvesstack$ factors
+through $\Curvesstack^{grc, 1}_g$,
+\item $X$ is geometrically reduced, geometrically connected,
+has dimension $1$, and has genus $g$.
+\end{enumerate}
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+The existence of the decomposition into open and closed substacks
+follows immediately from the discussion in Section \ref{section-open}
+and Lemma \ref{lemma-genus}. This proves the characterization in (1).
+The characterization in (2) follows as well since
+the genus of a geometrically reduced and connected
+proper $1$-dimensional scheme $X/k$ is defined
+(Algebraic Curves, Definition \ref{curves-definition-genus} and
+Varieties, Lemma
+\ref{varieties-lemma-proper-geometrically-reduced-global-sections})
+and is equal to $\dim_k H^1(X, \mathcal{O}_X)$.
+\end{proof}
+
+
+