# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Altman, Allen B.; Kleiman, Steven L., Compactifying the Picard scheme. II, Am. J. Math., 101, 1, 10-41, (1979), MR 527824 (81f:14025b) · Zbl 0427.14016 [2] Bayer, Dave; Eisenbud, David, Ribbons and their canonical embeddings, Trans. Am. Math. Soc., 347, 3, 719-756, (1995), MR 1273472 (95g:14032) · Zbl 0853.14016 [3] Bosch, Siegfried; Lütkebohmert, Werner; Raynaud, Michel, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 21, (1990), Springer-Verlag Berlin, MR 1045822 (91i:14034) · Zbl 0705.14001 [4] Chen, Dawei, Linear series on ribbons, Proc. Am. Math. Soc., 138, 11, 3797-3805, (2010), MR 2679602 (2011j:14071) · Zbl 1200.14058 [5] Donagi, Ron; Ein, Lawrence; Lazarsfeld, Robert, Nilpotent cones and sheaves on K3 surfaces, (Birational Algebraic Geometry, Baltimore, MD, 1996, Contemp. Math., vol. 207, (1997), Amer. Math. Soc. Providence, RI), 51-61, MR 1462924 (98f:14006) · Zbl 0907.32004 [6] Drézet, Jean-Marc, Faisceaux cohérents sur LES courbes multiples, Collect. Math., 57, 2, 121-171, (2006), MR 2223850 (2007b:14077) · Zbl 1106.14019 [7] Drézet, Jean-Marc, Moduli spaces of coherent sheaves on multiples curves, (Algebraic Cycles, Sheaves, Shtukas, and Moduli, Trends Math., (2008), Birkhäuser Basel), 33-43, MR 2402692 (2009e:14015) · Zbl 1222.14077 [8] Drézet, Jean-Marc, Faisceaux sans torsion et faisceaux quasi localement libres sur LES courbes multiples primitives, Math. Nachr., 282, 7, 919-952, (2009), MR 2541242 (2010j:14027) · Zbl 1171.14010 [9] Drézet, Jean-Marc, Sur LES conditions d’existence des faisceaux semi-stables sur LES courbes multiples primitives, Pac. J. Math., 249, 2, 291-319, (2011) · Zbl 1227.14016 [10] Eisenbud, David; Green, Mark, Clifford indices of ribbons, Trans. Am. Math. Soc., 347, 3, 757-765, (1995), MR 1273474 (95g:14033) · Zbl 0854.14016 [11] Hartshorne, Robin, Deformation theory, Graduate Texts in Mathematics, vol. 257, (2010), Springer New York, MR 2583634 (2011c:14023) · Zbl 1186.14004 [12] Inaba, Michi-Aki, On the moduli of stable sheaves on some nonreduced projective schemes, J. Algebr. Geom., 13, 1, 1-27, (2004), MR 2008714 (2004h:14020) · Zbl 1061.14013 [13] Kass, Jesse Leo, Two ways to degenerate the Jacobian are the same, Algebra Number Theory, 7, 2, 379-404, (2013), MR 3123643 · Zbl 1273.14061 [14] Langer, Adrian, Semistable sheaves in positive characteristic, Ann. Math. (2), 159, 1, 251-276, (2004), MR 2051393 (2005c:14021) · Zbl 1080.14014 [15] Lazarsfeld, Robert, Positivity for vector bundles, and multiplier ideals. II, (Positivity in Algebraic Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, A Series of Modern Surveys in Mathematics, vol. 49, (2004), Springer-Verlag Berlin), MR 2095472 (2005k:14001b) · Zbl 1093.14500 [16] Liu, Qing; Lorenzini, Dino; Raynaud, Michel, Néron models, Lie algebras, and reduction of curves of genus one, Invent. Math., 157, 3, 455-518, (2004), MR 2092767 (2005m:14039) · Zbl 1060.14037 [17] Le Potier, Joseph, Fibrés vectoriels sur LES courbes algébriques, Publications Mathématiques de l’Université Paris 7-Denis Diderot, vol. 35, (1995), Université Paris 7-Denis Diderot U.F.R de Mathématiques Paris, with a chapter by Christoph Sorger, MR 1370930 (97c:14034) · Zbl 0842.14025 [18] Maruyama, Masaki, Construction of moduli spaces of stable sheaves via Simpson’s idea, (Moduli of Vector Bundles, Sanda, 1994, Kyoto, 1994, Lecture Notes in Pure and Appl. Math., vol. 179, (1996), Dekker New York), 147-187, MR 1397986 (97h:14020) · Zbl 0885.14005 [19] Raynaud, M., Spécialisation du foncteur de Picard, Publ. Math. IHÉS, 38, 27-76, (1970), MR 0282993 (44 #227) · Zbl 0207.51602 [20] Simpson, Carlos T., Moduli of representations of the fundamental group of a smooth projective variety. I, Publ. Math. IHÉS, 79, 47-129, (1994), MR 1307297 (96e:14012) · Zbl 0891.14005 [21] Yang, Jin-Gen, Coherent sheaves on a fat curve, Jpn. J. Math. (N.S.), 29, 2, 315-333, (2003), MR 2035543 (2005c:14022) · Zbl 1084.14504
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.