## Calcul de quelques invariants des singularités de surface normale.(English)Zbl 0542.14001

Noeuds, tresses et singularités, C. R. Sémin., Plans-sur-Bex 1982, Monogr. Enseign. Math. 31, 191-203 (1983).
[For the entire collection see Zbl 0527.00017.]
Let X be a normal surface singularity over an algebraically closed field k and $$f:\tilde X\to X$$ be any resolution of the singular point $$x\in X$$. For an invertible sheaf $${\mathcal L}$$ on $$\tilde X$$ define $$-{\mathcal X}_ f({\mathcal L})=\dim_ kj_*j^*f_*({\mathcal L})/f_*({\mathcal L})+\dim_ kR^ 1f_*({\mathcal L})$$ where $$j:X-x\hookrightarrow X$$ is the inclusion. The author proves a relative Riemann-Roch formula, i.e. he computes $${\mathcal X}_ f({\mathcal L})-{\mathcal X}_ f({\mathcal O}_{\tilde X})-1/2c_ f({\mathcal L})(c_ f({\mathcal L})-c_ f({\mathcal K}))$$ in terms of invariants of a certain exceptional rational divisor, constructed from $${\mathcal L}$$. Here $$c_ f({\mathcal L})$$ denotes the exceptional rational divisor associated to $${\mathcal L}$$ and $${\mathcal K}$$ is the canonical sheaf. The invariants mentioned were introduced by Giraud and the author proves that these invariants can take only finitely many values for various $${\mathcal L}$$. Some applications to Hilbert-Samuel polynomials and an adjunction formula are proven.
Reviewer: G.-M.Greuel

### MSC:

 14B05 Singularities in algebraic geometry 14C40 Riemann-Roch theorems

### Keywords:

normal surface singularity; Riemann-Roch formula

Zbl 0527.00017