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Projective hypersurfaces with many singularities of prescribed types. (English) Zbl 1075.14034

Let \(\mathcal S\) be a set of (analytical or topological) types of isolated hypersurface singularities and for \(S_1, \ldots, S_r \in \mathcal S\) let \[ V_d^n(k_1S_1+\cdots + k_rS_r)=\left\{\begin{matrix} W\subset \mathbb P^n \text{ reduced hypersurface of degree } d \text{ with }\\ k_i \text{ singular points of type } S_i, i=1, \dots, r.\end{matrix} \right\} \] Hypersurfaces \(W\in V_d^n(k_1S_1+\cdots +k_r S_r)\) corresponding to \(T\)–smooth germs of the equisingular stratum are considered, i.e. the germ of the equisingular stratum is smooth and of the expected codimension. Assume the analytical singularity types are of co-rank less than \(n\) and the topological singularity types are of co-rank at most \(2\). If for some constant \(\alpha>0\) and all \(\{S_1, \ldots S_r\} \subset\mathcal S\) the condition \[ \sum_{i=1}^r k_i\tau^s(S_i)\leq \alpha d^n+O(d^{n-1}) \] implies the existence of a non-empty \(T\)-smooth component of \(V_d^n(k_1S_1+\cdots +k_r S_r)\), then it is called an asymptotically proper existence result for singularities of type \(\mathcal S\). Here \(\tau^s\) is the equianalytic or equisingular Tjurina number. Let \(\alpha_n (\mathcal S)\) be the supremum of all \(\alpha\) satisfying the property above. It is proved that if \(\tau^s (S)\) is bounded for \(S\in\mathcal S\) then \(\alpha_n(\mathcal S)\geq \frac{\alpha_{n-1}(\mathcal S)}{n}\). As a consequence one obtains \(\alpha_n(\mathcal S)\geq \frac{2}{9n!}\) for a set \(\mathcal S\) of singularity types of co-rank at most \(2\) with bounded Tjurina number. Note that \(\alpha_n(\mathcal S)\leq \frac{1}{n!}\). Furthermore, an asymptotically proper existence result is proved for hypersurfaces in \(\mathbb P^n\) with quasihomogeneous singularities.

MSC:

14J17 Singularities of surfaces or higher-dimensional varieties
14H20 Singularities of curves, local rings
14J70 Hypersurfaces and algebraic geometry

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