Néron models of algebraic curves (Q2796084)

The main result (Theorem 1.1) of this very informative and thorough paper is the following. Let \(S\) be a Dedekind scheme with function field \(K\), and let \(X_K\) be a proper smooth connected curve of strictly positive genus over \(K\). Let \(X\) be a minimal proper regular model of \(X_K\) over \(S\), and let \(X_{\text{sm}}\) be the smooth locus of \(X\). Then \(X_{\text{sm}}\) is a Néron model of \(X_K\). For more general curves, the authors also study under which condition Néron models or Néron lft-models exist.NEWLINENEWLINEThe idea of the proof is as follows. First a study is made of \(S\)-morphisms \(f : Y \to X\) with \(Y\) smooth and \(X\) a normal relative curve with smooth generic fiber. It is shown (Proposition 3.6) that if a fiber \(Y_s\) over a closed point \(s \in S\) is irreducible, then \(f (Y_s)\) is either a point or \(X\) is smooth at all points of this image. This implies that if \(X\) is an integral relative curve with smooth generic fiber, \(Y\) is smooth and irreducible, and \(f : Y \to X\) is an \(S\)-morphism with dominant generic fiber \(f_K : Y_K \to X_K\), then \(f\) factors through the minimal desingularization of \(X\).NEWLINENEWLINEThe authors then use this property and the closed immersion \(f : X_{K, \text{sm}} \to \text{Pic}_{X_K | K}^1\) to reduce the problem to the known case of Jacobians, thereby proving the main theorem. Note, however, that \(X_{\text{sm}}\) may not be faithfully flat over \(S\), nor need it embed into the Néron model of \(J_K\). Still, a finite morphism \(Y_K \to X_K\) from a proper smooth connected curve \(Y_K\) always extends to a morphism \(Y_{\text{sm}} \to X_{\text{sm}}\). For general smooth proper algebraic varieties \(X_K\) that admit a proper regular model \(X\) without rational curves in the fibers over elements of \(S\), it is also shown that \(X_{\text{sm}}\) is a Néron model of \(X_K\).NEWLINENEWLINEAfter studying the local case and the special case of conics, the authors show the following results (among others) for more general curves in the case where \(S\) is excellent:{\parindent=0.7cm\begin{itemize}\item[--] Let \(X_K\) be a proper regular (but not necessarily smooth) curve of strictly positive arithmetic genus over \(K\). Then the smooth locus of \(X_K\) admits a Néron model, which is constructed exactly as before. \item[--] (Theorem 7.10) Let \(U_K\) be an affine smooth connected curve over \(K\). Then \(U_K\) admits a Néron lft-model over \(S\) if \(U_K \ncong \mathbb{A}_L^1\) for any finite extension \(L\) of \(K\). \item[--] (Proposition 7.11) Let \(U_K\) be an affine smooth geometrically connected curve over \(K\), not isomorphic to \(\mathbb{A}_K^1\), and with regular compactification \(C_K\). Let \(\Delta_K = C_K \setminus U_K\). Let \(C\) be a relatively minimal regular model of \(C_K\) over \(S\), and let \(\Delta\) be the reduced Zariski closure of \(\Delta_K\) in \(C\). Then the Néron lft-model above is in fact a Néron model if and only if \(\Delta_K (K_s^{sh}) = \emptyset\) for all closed points \(s \in S\) and moreover \(\Delta_s \cap C_{\text{sh}, s} (k (s)^{\text{sep}}) = \emptyset\) for almost all \(s \in S\). In particular, the Néron lft-model is not a Néron model if \(K\) is of characteristic \(0\), \(S\) is infinite, and \(\Delta_K (K^{\text{sep}}) \neq \emptyset\). NEWLINENEWLINE\end{itemize}}