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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
##### Keywords:
ribbon; Simpson moduli space; generalised line bundles
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