Riemann-Roch theorem for normal surfaces and applications
type: Item
instance of: scholarly article
title: Riemann-Roch theorem for normal surfaces and applications
zbMATH Open document ID: 0877.14007
DOI: 10.1007/BF02953338
author: Yanyan Li
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1997-12-10T00:00:00Z
review text: In the theory of smooth compact complex surfaces, the Riemann-Roch formula \((\chi({\mathcal O}_X(D))= \chi(X)+\frac12 D(D- K_X))\) and the adjunction formula \((2p_a(C)-2= C^2+CK_X)\) are two of the most important foundational results. The purpose of this article is to work out these formulas in the case of normal projective surfaces and to show how they fit into various applications. Special emphasis is given to the case of projective orbisurfaces (i.e. surfaces with at most quotient singularities) since the results are much better in this more restricted setting. The basic problem which occurs in passing from smooth surfaces to normal surfaces is that there are, in general, Weil divisors on the surface which are not Cartier divisors. As a consequence, the intersection pairing has values in \(\mathbb{Q}\) instead of \(\mathbb{Z}\), and the RR (= Riemann Roch) formula and the adjunction formula need to be adjusted by correction terms. These correction terms can be localized at the finitely many singular points of \(X\). We now give a brief outline of the results in this article. The generalized RR theorem for Weil divisors on normal surfaces is given in (1.2). The correction terms (which turn out to be maps \(R_{X,x}: \text{Weil/Cart} (X,x)\to \mathbb{Q}\), where \(x\in \text{Sing }X\)) are calculated explicitly; they consist of an intersection theoretical contribution and a cohomological contribution. In (2.1) we state the generalized adjunction formula. The correction terms possess an additional symmetry, which stems from Serre duality. We use the previous mentioned results in order to give quick proofs of two results by Zariski: On a normal surface, the arithmetic genus \(p_a\) of curves can be naturally generalized to a map \(p_a: \text{Weil } X\to\mathbb{Z}\), and \(\chi({\mathcal O}_X(D))= \chi(X)-1+p_a(-D)\) for all \(D\in\text{Weil } X\). -- \S 4 contains two applications to embedded curves: Here again local correction terms appear, which are entirely expre…
Mathematics Subject Classification ID: 14C40
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 878389
zbMATH Keywords: Riemann-Roch formula
zbMATH Keywords: adjunction formula
zbMATH Keywords: orbisurfaces
zbMATH Keywords: quotient singularities
zbMATH Keywords: Weil divisors
zbMATH Keywords: Cartier divisors
zbMATH Keywords: second Chern class
zbMATH Keywords: globalizations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 878389