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**Effective behavior of multiple linear systems.**
*(English)*
Zbl 1092.14009

Let \(X\) be a smooth projective complex manifold. Then it is a fundamental problem to understand the behavior of a multiple linear system, and this problem has been studied by many authors. The purpose of this paper is to give effective versions of some well-known theorems on multiple linear systems for \(\dim X=2\). The main result of this paper is the following:

Let \(A\) be a nef and big divisor on a smooth projective complex surface \(X\), let \(T\) be any fixed divisor, and let \(k\) be a nonnegative integer. Assume that either \(n>k+\mathcal{M}(A,T)\), or \(n\geq \mathcal{M}(A,T)\) when \(k=0\) and \(T\sim K_{X}+\lambda A\) for some rational number \(\lambda\), where \(\mathcal{M}(A,T):=((K_{X}-T)A+2)^{2}/4A^{2}-(K_{X}-T)^{2}/4\). Suppose that there exists a zero dimensional subscheme \(\Delta\) on \(X\) with minimal degree \(\deg\Delta \leq k\) such that it does not give independent conditions on \(| nA+T| \). Then there is an effective divisor \(D\not=0\) containing \(\Delta\) such that \(TD-D^{2}-K_{X}D\leq k\) and \(DA=0\).

By direct applications of the above result for various \(T\), we obtain the effective version of known theorems.

Let \(A\) be a nef and big divisor on a smooth projective complex surface \(X\), let \(T\) be any fixed divisor, and let \(k\) be a nonnegative integer. Assume that either \(n>k+\mathcal{M}(A,T)\), or \(n\geq \mathcal{M}(A,T)\) when \(k=0\) and \(T\sim K_{X}+\lambda A\) for some rational number \(\lambda\), where \(\mathcal{M}(A,T):=((K_{X}-T)A+2)^{2}/4A^{2}-(K_{X}-T)^{2}/4\). Suppose that there exists a zero dimensional subscheme \(\Delta\) on \(X\) with minimal degree \(\deg\Delta \leq k\) such that it does not give independent conditions on \(| nA+T| \). Then there is an effective divisor \(D\not=0\) containing \(\Delta\) such that \(TD-D^{2}-K_{X}D\leq k\) and \(DA=0\).

By direct applications of the above result for various \(T\), we obtain the effective version of known theorems.

Reviewer: Yoshiaki Fukuma (Kochi)

### MSC:

14C20 | Divisors, linear systems, invertible sheaves |

14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |

14C40 | Riemann-Roch theorems |

14J99 | Surfaces and higher-dimensional varieties |