Curve singularities arising from the representation theory of tame
hereditary algebras
by Helmut Lenzing. Submitted 11-20-1984. Appeared 1987.
Geometry. As for stacks, it is not easy to give a short, down to earth, geometric description of a weighted projective line. Roughly speaking a weighted projective line X is a projective line with a finite number of weird points. The weirdness of each of those special points is measured by an integer >2 called its weight. That's the source of the terminology. The weird points are stacky points, and if the weight of the point is n the stacky structure at that point is BZ_n, the classifying stack for the cyclic group of order n.
There is a morphism f:X--->P^1 that is an isomorpism away from the stacky points. I should say what category this is in but I won't.
As for a scheme, there is an abelian category Qcoh(X). The morphism f induces, or is defined by, an adjoint pair of functors, the direct and inverse image functors f_*:Qcoh(X)-->Qcoh(P^1) and f^*:Qcoh(P^1)--->Qcoh(X).
The functor f_* is faithful and has a right adjoint, f^!, so f behaves like an affine morphism. In particular, as is the case for an affine morphism of schemes, there is a coherent sheaf of algebras A on P^1 and Qcoh(X) is equivalent to the category of quasi-coherent A-modules.
The justification for calling X is a stacky projective line is that there is a Deligne-Mumford stack whose category of quasi-coherent sheaves is equivalent to Qcoh(X) (even as monoidal categories), and whose coarse moduli space is P^1. Indeed, f_* and f^* also agree with the usual functors to and from the quasi-coherent sheaves on the coarse moduli space.
If Y is a projective algebraic curve, there is a commutative Z-graded algebra S of Krull dimension two such that Y=Proj(S). A weighted projective line also has a homogeneous coordinate ring that is a finitely generated commutative algebra of Krull dimension twa. A crucial difference is that the S for a weighted projective line X is graded by an abelian group of rank one that may have torsion and, even if the grading group is Z it will not be possible to place all the generators of S in degree one. For example, S might be k[x,y,z]/(x^2+y^3+z^4) with deg(x,y,z)=(6,4,3). I hasten to add that X is not Proj S.
There is a functor from the category of graded S-modules to the category Qcoh(X). Rather than using sheaves, as one does in algebraic geometry, we will tend to work directly with graded S-modules.
As for algebraic curves, we wish to understand Qcoh(X). If Y is an irreducible (we only need reduced) algebraic curve then every object in coh(Y) is a direct sum of a finite length object F and an object V that has no finite length subobjects. If Y is smooth and irreducible, V is locally free, i.e., a vector bundle. Furthermore, F decomposes into a direct sum of smaller objects according to its support; i.e., F is the direct sum of objects F_y, where F_y is the part of F supported at the point y in Y. Each F_y is a direct sum of indecomposable objects each of which looks like k[t]/(t^n) for some n.
The same phenomenon occurs for a weighted projective line, X. The noetherian objects in Qcoh(X) form an abelian subcategory coh(X) and every object in coh(X) is a direct sum F+V where F has finite length and V has no finite length subobjects. There is a further decomposition of F as a direct sum of various objects F_x but and each F_x is a sum of indecomposable objects, but now, in contrast to the case of an algebraic curve, those indecomposables can be more complicated than k[t]/(t^n). Still, the indecomposable objects of finite length in Qcoh(X) can be completely described and the real problem arrives, as it does for smooth algebraic curves: describe all possible V, the ``vector bundles''.
The complexity of the vector bundles on a curve rises with the genus of Y. If Y has genus zero it is the projective line and every V is a direct sum of O(n)s, twists of the structure sheaf. In particular, the only indecomposable bundles are the line bundles. When the genus is 1, Y is an elliptic curve and there are now indecomposable vector bundles of arbitrarily large rank. These do, however, admit a complete classification, found by Atiyah as a graduate student. Once the genus exceeds 1, it is a hopeless problem to classify all vector bundles, but there is still plenty to say about various moduli spaces of vector bundles.
A weighted projective line has a genus, but it need not be an integer. The complexity of the classification of the Vs again is governed by the genus.
I have run out of steam and this is already too long! But I haven't said anything about the algebra. Weighted projective lines first arose in the representation theory of finite dimensional algebras.
Algebra. If X is a weighted projective line, there is a finite dimensional algebra A such that the bounded derived categories D^b(coh(X)) and D^b(mod(A)) are equivalent. The algebra A is the endomorphism ring of a tilting bundle on X (we will describe what this means) and X arises from A as a sort of moduli space for certain representations of A. I should say more but I'm tired, and you might be too!
Google gives about 14,600 hits when "weighted projective lines'' is entered, so where to begin?
Early papers.
Curve singularities arising from the representation theory of tame
hereditary algebras
by Helmut Lenzing. Submitted 11-20-1984. Appeared 1987.
A class of weighted projective curves arising in representation
theory of finite dimensional algebras
by Werner Geigle and Helmut Lenzing. Appeared 1987. Top hit on
Google Scholar, cited by 199.
The preprojective algebra of a tame hereditary artin algebra,
by Dagmar Baer, Werner Geigle and Helmut Lenzing. Submitted July
1986. Appeared 1987
Surveys.
Representations of finite-dimensional algebras and singularity
theory
by Helmut Lenzing,
in: Trends in ring theory (Miskolc, 1996), 71-97,
Canadian Math. Soc. Conf. Proc., 22, Amer. Math.
Soc., Providence, RI, 1998.
Introduction to coherent sheaves on weighted projective lines,
by Xiao-Wu Chen and Henning Krause.
arXiv:math/0911.4473v3.
Memoirs. There are two longish accounts of weighted projective lines. Neither aims to be comprehensive. Each covers a number of aspects in detail.
Hagen Meltzer, Exceptional vector bundles, tilting sheaves and
tilting complexes for weighted projective lines,
Mem. Amer. Math.
Soc. 171 (2004), no. 808.
Dirk Kussin, Noncommutative curves of genus zero: related to finite
dimensional algebras,
Mem. Amer. Math. Soc. 201 (2009), no. 942.
Summary of topics. There have been many seminars and mini-courses similar to our seminar. Here are handouts for two such.
Hefei lectures: Weighted projective lines and applications
by Helmut Lenzing
Bielefeld Seminar
Speakers. We need volunteers.
October 7. Paul Smith. Introduction and notation.
October 14. Paul Smith. The basics.
October 21.
October 28.
November 4. Chelsea Walton. Quotient categories, especially QGr(S),
and Serre's Theorem QGr(S) = Qcoh(Proj S) when S is graded by the
integers and generated over the base field by its degree 1
component. (This will never be the case for the S that concern us,
but it does explain why QGr(-) is a natural construction that takes
us from graded modules to something like a category of
quasi-coherent sheaves.