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Hypersurfaces with defect. (English) Zbl 1469.14009

The notion of defect was introduced as a measure to what extend the imposition of double points fails to lower the dimension of certain cohomology groups [C. H. Clemens, Adv. Math. 47, 107–230 (1983; Zbl 0509.14045)].
In the article under review, a hypersurface \(X\) in \(\mathbb{P}^n\) is said to have defect, if \(h^i(X)\neq h^i(\mathbb{P}^n) \) for some \(i\in\{n,\dots,2n-2\}\), where \(h^i\) stands for the \(i\)-th Betti number in a “reasonable” cohomology theory.
Extending the classical situation (projective hypersurfaces over the complex numbers with only ordinary double points), the author allows base fields of arbitrary characteristic and hypersurfaces with more general singularities. Results are given in the spirit of “defect implies many singularities”.
In Theorem 1.1. an estimate is obtained for the global Tjurina number; the result is non-trivial if all singularities of \(X\) are isolated:
Let \(K\) be a field of characteristic zero. Suppose that \(X\subseteq\mathbb{P}^n_K\), \(n\geq 3\), is a hypersurface with defect in algebraic de Rham, Kähler-de Rham, singular or étale cohomology. Denote by \(\tau (X)\) the global Tjurina number of \(X\). Then \(\tau (X)\geq\frac{\mathrm{deg}(X)-n+1}{n^2+n+1}\).
Moreover, if \(X\) has at most weighted homogeneous singularities, then \(\tau(X)\geq\mathrm{deg}(X)-n+1\).
If the base field has positive characteristic, the list of simple singularities is found in [G. M. Greuel and H. Kröning, Math. Z. 203, No. 2, 339–354 (1990; Zbl 0715.14001)]; they are known to be absolutely isolated with relatively easy resolutions, especially for the \(A_k\)-type-singularities.
Based on a detailed description of the resolutions, the author states the following for a base field \(K\) of characteristic \(\neq 2\) and if the hypersurface \(X\subseteq\mathbb{P}_K^n\) has defect with respect to étale or rigid cohomology:
Assume \(X\) has only singularities \(x\in X\) which are ordinary multiple points of multiplicity \(m_x\) or of type \(A_{k_x}\), then \(\sum m_x+\sum 2\left\lceil \frac{k_x}{2}\right\rceil\geq\mathrm{deg}(X)\).
Reviewers remark: Local resolution graphs of \(A\)-type singularities coincide in all characteristics. Furthermore, they are weighted homogeneous for \(n\) even. This might indicate a possible extension of the result to the case \(\mathrm{char }(K)=2\).

MSC:

14B05 Singularities in algebraic geometry
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14F20 Étale and other Grothendieck topologies and (co)homologies
14F42 Motivic cohomology; motivic homotopy theory
14J17 Singularities of surfaces or higher-dimensional varieties
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References:

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