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Moduli of generalized line bundles on a ribbon. (English) Zbl 1372.14018
A ribbon \(X\) is a first order thickening of a nonsingular curve. More precisely, \(X\) is an irreducible projective variety of dimension \(1\) defined over an algebraically closed field \(k\) such that the reduced subscheme \(X_{\mathrm{red}}\) of \(X\) is a nonsingular curve and the nilradical \(N \subset \mathcal{O}_X\) is locally generated by a nonzero element whose square is \(0\). Let \(g := 1- \chi(\mathcal{O}_X)\) be the genus of \(X\) and \(\bar{g}\) the genus of \(X_{\mathrm{red}}\). For a polarised ribbon \((X, L)\), the authors study the Simpson moduli space \(M(X, P_d)\) of semistable sheaves on \(X\) with Hilbert polynomial \(P_d(t) = deg(L) t + d+1-g\) and its open subset \(M^s(X,P_d)\) consisting of stable sheaves. A generalised line bundle \(I\) on \(X\) is a coherent sheaf on \(X\) of the form \(I = f_*I'\) where \(f: X' \to X\) is a finite birational morphism of ribbons. The degree of \(I\) is defined as \(\deg(I)= \chi(I) - \chi(\mathcal{O}_X)\) and index of \(I\) is \(b(I)= g - g(X')\). The moduli space \(M(X,P_d)\) (resp. \(M^s(X,P_d)\)) parametrises two types of objects, namely direct images of slope semistable (resp. slope stable) vector bundles on \(X_{\mathrm{red}}\) of rank \(2\), degree \(d-1-g+2\bar{g}\) and generalised line bundles on \(X\) of degree \(d\) and index at most \(1+g -2\bar{g}\) (resp. strictly less than \(1+g- 2\bar{g}\)). They prove several results on the geometry of \(M(X,P_d)\). The moduli space \(M(X,P_d)\) is connected. If there exists a generalised line bundle (i.e., \(g>2\bar{g}\)), then \(M(X,P_d)\) has \(n\) irreducible components each of dimension \(g\), with \(n= |(g+2)/2| - \bar{g}\) if \(d\) is even and \(n= |(g+1)/2| -\bar{g}\) if \(d\) is odd, here \(|m|\) denotes the integral part of \(m\). There is at most one additional component. If it exists, it is of dimension \(4\bar{g}-3\) and has general element coming from a stable vector bundle of rank \(2\) on \(X_{\mathrm{red}}\). It exists if \(\bar{g} \geq 2\) and \(4\bar{g}-3 \geq g\), it does not exist if \(\bar{g}=0\) or \(1\). If \(\bar{g}\geq 2\) and \(g > 4\bar{g}-3\), then the smooth locus of \(M(X,P_d)\) consists of line bundles on \(X\).
Generalised line bundles (and generalised linear series) were defined by D. Eisenbud and M. Green [Trans. Am. Math. Soc. 347, No. 3, 757–765 (1995; Zbl 0854.14016)]. The case \(\bar{g}=0\) was studied in detail by Yang. Drézet has several papers on slope semistable sheaves on a multiple curve. Inaba has studied stable sheaves on a non-reduced scheme.

MSC:
14H10 Families, moduli of curves (algebraic)
14D20 Algebraic moduli problems, moduli of vector bundles
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