These are notes for a minicourse on Hall algebras given at the ICTP in Trieste in January 2006. After giving the definition and first properties of Hall algebras, we study in some details the classical Hall algebra, the Hall algebra of quivers, and the Hall algebra of coherent sheaves on smooth projective curves. The last section deals with the Hall algebras in the context of derived categories. (The figures only seem to come out nicely in the .ps file)

/pdf/lectures-on-hall-algebras-4nkdeafypj.pdf

Lectures on hall algebras

In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show that the dimension of the bounded derived category of coherent sheaves on a rational projective curve with only nodal or cuspidal singularities is at most two. In the case of the Kodaira cycles of projective lines, the corresponding tilted algebras belong to a well-known class of gentle algebras. We work out in details the tilting equivalence in the case of the Weierstrass nodal curve zy2 = x3 + x2z.

/pdf/tilting-on-non-commutative-rational-projective-curves-4bkubthqgv.pdf

Tilting on non-commutative rational projective curves

We show that all reduced closed subschemes of projective space that have a Cohen-Macaulay graded coordinate ring are of wild Cohen-Macaulay type, except for a few cases which we completely classify.

/pdf/the-cohen-macaulay-representation-type-of-arithmetically-25arm7sd3i.pdf

The Cohen-Macaulay representation type of arithmetically Cohen-Macaulay varieties

We establish a $\mathbb{Z}[[t_1,\ldots, t_n]]$-linear derived equivalence between the relative Fukaya category of the 2-torus with $n$ distinct marked points and the derived category of perfect complexes on the $n$-Tate curve. Specialising to $t_1= \ldots =t_n=0$ gives a $\mathbb{Z}$-linear derived equivalence between the Fukaya category of the $n$-punctured torus and the derived category of perfect complexes on the standard (Neron) $n$-gon. We prove that this equivalence extends to a $\mathbb{Z}$-linear derived equivalence between the wrapped Fukaya category of the $n$-punctured torus and the derived category of coherent sheaves on the standard $n$-gon.

Arithmetic mirror symmetry for genus 1 curves with $n$ marked points

In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show the dimension of the bounded derived category of coherent sheaves on a rational projective curve with only nodal or cuspidal singularities is at most two. In the case of the Kodaira cycles of projective lines, the corresponding tilted algebras belong to a well-known class of gentle algebras. We work out in details the tilting equivalence in the case of the plane nodal cubic.

We study the derived categories of coherent sheaves on some singular projective curves and give a complete description of indecomposable objects using the technique of matrix problems.

/pdf/coherent-sheaves-on-rational-curves-with-simple-double-2tqip80a6u.pdf

Coherent sheaves on rational curves with simple double points and transversal intersections

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Coherent sheaves on rational curves with simple double points and transversal intersections