We study the Variations of mixed Hodge structures (VMHS) associated to a pencil ${\cal X}$ (parametrised by an open set $B \subset {\Bbb P}^1$) of equisingular hypersurfaces of degree $d$ in ${\Bbb P}^{4}$ with exactly $m$ ordinary double points as singularities as well as the variations of Hodge structures (VHS) associated to the desingularization of this family $ \widetilde{\cal X}$. The case where exactly $l \le m $ of those double points are in algebraic general position (short:agp) is studied in detail and determine the possible limiting mixed Hodge structures (LMHS) associated to each of the points in ${\Bbb P}^1\backslash B$. We find that the position of the singular points being in agp is not sufficient to describe the space of first one-adjoint conditions and naturally the notion of a set of singular points being in homologically good position (short: hg) is introduced. By requiring that the set of nodes in agp is also in hg, the $F^2$-term of the Hodge filtration of the desingularization is completely determined. The particular pencil $ {\cal X}$ of quintic hypersurfaces with $100$ singular double points with $86$ of them in agp which served as the starting point for this paper is treated with particular attention.

中文翻译：

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与 Calabi-Yau 三元组的等价一维族相关的混合 Hodge 结构的变化

我们研究了与铅笔 ${\cal X}$（由开集 $B \subset {\Bbb P}^1$ 参数化）相关的混合霍奇结构（VMHS）的变化${\Bbb P}^{4}$ 正好有 $m$ 普通双点作为奇点，以及与该族 $\widetilde{\cal X}$ 的去奇异化相关的霍奇结构 (VHS) 的变化。详细研究了这些双点中的 $l \le m $ 正好处于代数一般位置 (short:agp) 的情况，并确定与 ${\Bbb 中的每个点相关联的可能的极限混合霍奇结构 (LMHS) P}^1\反斜杠 B$。我们发现奇异点在 agp 中的位置不足以描述第一个单伴随条件的空间，自然而然地引入了一组处于同调好位置（简称：hg）的奇异点的概念。通过要求 agp 中的节点集也在 hg 中，完全确定了去奇异化的 Hodge 过滤的 $F^2$-term。特别关注具有 $100$ 奇异双点的五次超曲面的特定铅笔 ${\cal X}$，其中 $86$ 在 agp 中，作为本文的起点，受到了特别关注。