** 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.**

**Surveys.**

**Memoirs.**
There are two longish accounts of weighted projective lines.
Neither aims to be comprehensive. Each covers a number of aspects
in detail.

**Summary of topics.**
There have been many seminars and mini-courses
similar to our seminar. Here are handouts for two such.

**Speakers.**
We need volunteers.