Defect of a nodal hypersurface. (English) Zbl 0983.14017
In this paper, the author gives interesting formulas for the Hodge numbers of a nodal hypersurface in a smooth complex projective fourfold. Let \(X\) be a smooth complex projective fourfold and let \(Y\) be a nodal hypersurface in \(X\) such that:
A1: the line bundle \({\mathcal M}:={\mathcal O}_X(Y)\) is ample,
A2: \(H^2\Omega^1_X=0,\)
A3: \(H^3(\Omega^1_X \otimes{\mathcal M}^{-1}) =0\).
Denote by \(\widetilde Y\) a big resolution of \(Y\) and \(\widehat Y\) a small one.
Theorem 1: \[ \begin{aligned} & h^{11}(\widetilde Y)=h^{11}(X) +\mu+\delta,\\ & h^{12}(\widetilde Y)=h^0({\mathcal M}^{\otimes 2}\otimes K_X)+ h^3{\mathcal O}_x-h^0( {\mathcal M}\otimes K_X)-h^3\Omega^1_X-h^4(\Omega^1_X\otimes{\mathcal M}^{-1})-\mu +\delta\end{aligned} \] Theorem 2: \[ h^{11}(\widehat Y)=h^{11}(\widetilde Y)-\mu,\quad h^{12} (\widehat Y)=h^{12}(\widetilde Y), \] where \(\mu\) is the number of nodes and \(\delta\) the defect of \(Y\).
A1: the line bundle \({\mathcal M}:={\mathcal O}_X(Y)\) is ample,
A2: \(H^2\Omega^1_X=0,\)
A3: \(H^3(\Omega^1_X \otimes{\mathcal M}^{-1}) =0\).
Denote by \(\widetilde Y\) a big resolution of \(Y\) and \(\widehat Y\) a small one.
Theorem 1: \[ \begin{aligned} & h^{11}(\widetilde Y)=h^{11}(X) +\mu+\delta,\\ & h^{12}(\widetilde Y)=h^0({\mathcal M}^{\otimes 2}\otimes K_X)+ h^3{\mathcal O}_x-h^0( {\mathcal M}\otimes K_X)-h^3\Omega^1_X-h^4(\Omega^1_X\otimes{\mathcal M}^{-1})-\mu +\delta\end{aligned} \] Theorem 2: \[ h^{11}(\widehat Y)=h^{11}(\widetilde Y)-\mu,\quad h^{12} (\widehat Y)=h^{12}(\widetilde Y), \] where \(\mu\) is the number of nodes and \(\delta\) the defect of \(Y\).
Reviewer: S.Ohyanagi (Fujisawa)
MSC:
| 14J17 | Singularities of surfaces or higher-dimensional varieties |
| 14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |
| 14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |
| 14B05 | Singularities in algebraic geometry |
| 14J30 | \(3\)-folds |
