Defect of projective hypersurfaces with isolated singularities
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915733770862592 |
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| author | Jung, Seung-Jo Saito, Morihiko |
| author_facet | Jung, Seung-Jo Saito, Morihiko |
| contents | Let $X$ be a hypersurface with isolated singularities defined by $f$ in ${\bf P^{n+1}}$ with $n>1$. The difference ${\rm def}(X):=h^{n+1}(X)-h^{n-1}(X)$ is called the defect of $X$ (for self-duality of the cohomology of $X$). It is known that its vanishing is closely related to ${\bf Q}$-factoriality of $X$ in the rational singularity case with $n=3$. This number coincides with the dimension of the cokernel of the inclusion $H^{n-1}(X)\to{\rm IH}^{n-1}(X)$, the rank of the morphism from the vanishing cohomologies of $X$ to $H^{n+1}(X)$ for a one-parameter smoothing of $X$ with total space smooth, and also with the dimension of the unipotent monodromy part of the Milnor fiber cohomology of $f$ with degree $n$. In the case $X$ has only weighted homogeneous isolated singularities, the defect ${\rm def}(X)$ is then given by the $E_2$-term of the spectral sequence of the double complex with differentials ${\rm d}f\wedge$ and $\rm d$ by the $E_2$-degeneration of the pole order spectral sequence. It can be calculated explicitly using a computer even for analogues of the Hirzebruch quintic threefold with more than one hundred ordinary double points found by B.\ van Geemen and J.\ Werner in a compatible way with their computation. We give also an example with ${\rm def}(X)>0$ and $|{\rm Sing}\,X|=1$ where $n=3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23522 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Defect of projective hypersurfaces with isolated singularities Jung, Seung-Jo Saito, Morihiko Algebraic Geometry Let $X$ be a hypersurface with isolated singularities defined by $f$ in ${\bf P^{n+1}}$ with $n>1$. The difference ${\rm def}(X):=h^{n+1}(X)-h^{n-1}(X)$ is called the defect of $X$ (for self-duality of the cohomology of $X$). It is known that its vanishing is closely related to ${\bf Q}$-factoriality of $X$ in the rational singularity case with $n=3$. This number coincides with the dimension of the cokernel of the inclusion $H^{n-1}(X)\to{\rm IH}^{n-1}(X)$, the rank of the morphism from the vanishing cohomologies of $X$ to $H^{n+1}(X)$ for a one-parameter smoothing of $X$ with total space smooth, and also with the dimension of the unipotent monodromy part of the Milnor fiber cohomology of $f$ with degree $n$. In the case $X$ has only weighted homogeneous isolated singularities, the defect ${\rm def}(X)$ is then given by the $E_2$-term of the spectral sequence of the double complex with differentials ${\rm d}f\wedge$ and $\rm d$ by the $E_2$-degeneration of the pole order spectral sequence. It can be calculated explicitly using a computer even for analogues of the Hirzebruch quintic threefold with more than one hundred ordinary double points found by B.\ van Geemen and J.\ Werner in a compatible way with their computation. We give also an example with ${\rm def}(X)>0$ and $|{\rm Sing}\,X|=1$ where $n=3$. |
| title | Defect of projective hypersurfaces with isolated singularities |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2512.23522 |