×

Torsion-free rank one sheaves over del Pezzo orders. (English) Zbl 1376.14046

Let \(e,e'\) and \(f\) be positive integers with \(e'\) dividing \(e\) and let \(R:= \mathbb{C}[[u,v]]\) where \(\mathbb{C}\) is the field of complex numbers. Let \(S:= R<x,y>/I\) where \(I\) is the ideal generated by \(x^{e'}-u, y^{e'}-v, yx- \beta xy\) where \(\beta\) is a primitive \(e'\)’th root of unity. It follows \(S\) is of finite rank as free left \(R\)-module. The authors define a sub-\(R\)-algebra \(B\) of the full matrix ring \(\mathrm{Mat}^{e/e',e/e'}(S)\). And defines \(A(e,e',f)\) to be the full matrix ring \(\mathrm{Mat}^{f,f}(B)\).
Note: It is not immediate that \(\mathrm{Mat}^{f,f}(B)\) is an associative ring. Let \(X\) be a smooth projective surface over \(\mathbb{C}\) -the field of complex numbers. The authors define an “order” on \(X\) to be a sheaf of associative \(O_X\)-algebras \(A\) satisfying the following properties:
1.
\(A\) is coherent and torsion free as left \(O_X\)-module.
2.
The stalk \(A_{\eta}\) at the generic point is a central division algebra over \(\mathbb{C}(x):=O_{X,\eta}\) – the function field of \(X\).
The authors claim that the “set of orders in \(A_{\eta}\)” has an ordering defined by inclusion and defines a “maximal order on \(X\)” to be an order on \(X\) that is maximal with respect to this ordering. The authors define a maximal order \(A\) on a smooth projective surface \(X\) over \(\mathbb{C}\) to be “terminal” if the following conditions hold: Let \(x \in X\) be a closed point.
1.
There is an isomorphism \(\tilde{O}_{X,x} \cong \mathbb{C}[[u,v]]\) of rings.
2.
There is a \(\mathbb{C}[[u,v]]\)-algebra isomorphism \(A_x \otimes \tilde{O}_{X,x} \cong A(e,e',f)\) for some positive integers \(e,e',f\) where \(e'\) divides \(e\).
Note: Since \(X\) is assumed to be smooth the condition 1 automatically satisfied.
The main theorem of the paper is the following:
Theorem. Let \(X\) be a smooth projective surface over \(\mathbb{C}\). Let \(A \neq O_{\mathbb{P}^2}\) be a terminal order on \(X\) with \(-K_A\) ample. Then every torsion free \(A\)-module \(E\) can be deformed to a locally free \(A\)-module \(E'\). They use this result to prove that the space \(M^{lf}_{A/\mathbb{P}^2;c_1,c_2}\) of locally free \(A\)-modules with Chern classes \(c_1,c_2\) is a dense open subset of \(M_{A/ \mathbb{P}^2,c_1,c_2}\).

MSC:

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14D15 Formal methods and deformations in algebraic geometry
16H10 Orders in separable algebras
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Artamkin, Igor, Deforming torsion-free sheaves on an algebraic surface, Math. USSR, Izv., 36, 3, 449-486 (1991) · Zbl 0723.14011
[2] Chan, Daniel; Chan, Kenneth, Rational curves and ruled orders on surfaces, J. Algebra, 435, 52-87 (2015) · Zbl 1344.14003
[3] Chan, Daniel; Ingalls, Colin, The minimal model program for orders over surfaces, Invent. Math., 161, 427-452 (2005) · Zbl 1078.14005
[4] Chan, Daniel; Kulkarni, Rajesh, Del Pezzo orders on surfaces, Adv. Math., 173, 144-177 (2003) · Zbl 1051.14005
[5] Ellingsrud, Geir; Lehn, Manfred, Irreducibility of the punctual quotient scheme of a surface, Ark. Mat., 37, 2, 245-254 (1999) · Zbl 1029.14017
[6] Hoffmann, Norbert; Stuhler, Ulrich, Moduli schemes of generically simple Azumaya modules, Doc. Math., 10, 369-389 (2005) · Zbl 1092.14051
[7] Matsumura, Hideyuki, Commutative Ring Theory (1989), Cambridge University Press · Zbl 0666.13002
[8] Reede, Fabian, Moduli Spaces of Bundles over Two-Dimensional Orders (2013), Mathematisches Institut der Georg-August-Universitaet: Mathematisches Institut der Georg-August-Universitaet Goettingen, PhD thesis · Zbl 1285.14001
[9] Tannenbaum, Allen, The Brauer group and unirationality: an example of Artin-Mumford, (Kervaire, Michel; Ojanguren, Manuel, Groupe de Brauer (1981), Springer-Verlag), 103-128 · Zbl 0453.13003
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.