on chern classes and Riemann Roch theorem for torsion-free sheaves on singular (possibly multiple) curve

Question

1. I'm looking for a definition of Chern class (at least the first one) for a torsion-free sheaf $F$ (not necessarily locally free) on a singular curve (for simplicity can assume all the singularities are planar).

The Chern class can be, of course, extracted from an exact sequence relating $F$ to some locally free sheaves. But I would like some more direct definition, like the one given by Hartshorne (Generalized divisors on Gorenstein curves and a theorem of Noether. J. Math. Kyoto Univ. 26 (1986), no. 3, 375--386).

At least for the first Chern class.

1.5 Even if one wants to define $c_1(F)$ from some resolution: torsion free sheaves on singular curves sometimes have no finite locally free resolutions. What would you do in this case?

1. I'm looking for the Riemann-Roch for torsion-free sheaves on a singular curve (can assume the singularities to be planar). For example Hartshorne in the paper above does it for rank one.

Of course, if the only definition of the first Chern class is from the exact sequence, then Riemann-Roch is tautological (an alternative way to define $c_1(F)$). So this question is meaningful modulo the first question.

Somehow I do not find all this in classical textbooks.

Thanks to everybody!!!!

Answer 1

In the affine case, there is a sweet way to define the first Chern class as follows:

Let $R$ by the coordinate ring and $M$ the $R$-module correspond to our sheaf. As $M$ is torsion-free, one can embed $M$ in to a free module: $ 0\to M \to F \to N \to 0$ ( you need $M$ to be of constant rank, and that rank would be the rank of $F$). In this $N$ would be torsion, so the support is finite. Take the cycle $c(N) = \sum length(N_p)[p]$ where $p$ runs over the support of $N$. Then define $c(M)= -c(N)$.

In general, one could get codimension 1 cycles by picking them from any prime filtration of $M$(you needs to show that what you get from 2 different filtrations are rationally equivalent). This paper contains a treatment of that result.

Answer 2

Well, in this paper, there seems to be a definition for curves with nodes. It seems to go like this: Let $\mathcal{F}$ be a torsion free sheaf of rank $r$, then at each node $x_i$, there's a decomposition $\mathcal{F}_{x_1}=\mathcal{O}_{x_i}^{a_i}\oplus\mathfrak{m}_{x_i}^{r-a_i}$. At each point, we also get a map $\mathcal{F}_{x_i}\to k(x_i)^{a_i}$, and so we get $$0\to\mathcal{E}\to\mathcal{F}\to\oplus_{x_i} k(x_i)^{a_i}\to 0$$. Then, there's a vector bundle $E$ on the resolution, and we get the formula $\deg\mathcal{F}(=c_1)=\deg E-nr+\sum a_i$ where $n$ is the number of nodes.

Admittedly, this is a bit awkward, and only works in the case of nodes, but it seems a place to start...I don't know how much of this still remains true for planar singularities, or how the formulas would change.

Answer 3

I must be missing something here. Why not use the standard Baum-Fulton-MacPherson $\tau$ map? This is the map one needs for Riemann-Roch to work anyway as in sections 18.2 and 18.3 of Fulton's "Intersection theory" book. The definition is very natural - embed the curve in a smooth variety (e.g. in projective space) and resolve the push-forward by vector bundles on the target. You only need to be careful and correct your computation by intersecting with the Todd class of the target. Baum-Fulton-MacPherson check that the definition is independent of the embedding and that Riemann-Roch holds for l.c.i. proper morphisms, e.g. for projective curves with planar singularities.

If we want to cheat we can also define the degree of a torsion free sheaf $F$ on a curve $C$ from the Hilbert polynomial, or simply as $\deg(F) = \chi(F) - rk(F)\chi(\mathcal{O}_C)$.