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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

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