On the fixed loci of the canonical systems over normal surface singularities (Q1018239)

Let \((V,0)\) be the germ of a normal surface singularity and \(\pi:X\to V\) its minimal resolution. Let \(L\) be a line bundle on \(X\) and \(H^0(X,L)\neq 0\). The fixed part of the linear system \(|L|\) is the biggest effective divisor \(F\) supported in \(\pi^{-1}(0)\) such that the restriction map \(H^0(X,L)\to H^0(F,L)\) is zero. Assume \(L-K_X\) is nef. It is proved that the fixed part of \(|L|\) supports at most exceptional sets of rational singular points. If \((U,p)\) denotes the rational singular point obtained by contracting a connected component of the fixed part of \(|L|\) then the multiplicity \(\text{mult}(U,p)\) and the embedding dimension satisfy \(\text{mult}(U,p)\leq 2_{p_f}(V,0)\) and \(\text{embdim}(U,p)\leq 2_{p_f}(V,0)+1\). Here \(p_f(V,0)\) is the arithmetic genus of the fundamental cycle on \(\pi^{-1}(0)\).