On singularities of determinantal hypersurfaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911407388229632 |
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| author | Bath, Daniel Mustaţă, Mircea |
| author_facet | Bath, Daniel Mustaţă, Mircea |
| contents | Given a closed subscheme $Z$ in a smooth variety $X$, defined by the maximal minors of an $s\times r$ matrix of regular functions, with $s\geq r$, we consider the corresponding incidence correspondence $W$ in $Y=X\times {\mathbf P}^{r-1}$, and relate the log canonical thresholds of $(X,Z)$ and $(Y,W)$. In particular, when $r=s$, we show that ${\rm lct}(X,Z)=1$ if and only if ${\rm lct}(Y,W)=r$. Moreover, in this case, we show that $Z$ has rational singularities if and only if $W$ has pure codimension $r$ in $Y$ and has rational singularities. As a consequence, we deduce that for a configuration hypersurface with a connected configuration matroid, the corresponding configuration incidence variety has rational singularities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_22072 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On singularities of determinantal hypersurfaces Bath, Daniel Mustaţă, Mircea Algebraic Geometry 14B05, 14J17, 32S25 Given a closed subscheme $Z$ in a smooth variety $X$, defined by the maximal minors of an $s\times r$ matrix of regular functions, with $s\geq r$, we consider the corresponding incidence correspondence $W$ in $Y=X\times {\mathbf P}^{r-1}$, and relate the log canonical thresholds of $(X,Z)$ and $(Y,W)$. In particular, when $r=s$, we show that ${\rm lct}(X,Z)=1$ if and only if ${\rm lct}(Y,W)=r$. Moreover, in this case, we show that $Z$ has rational singularities if and only if $W$ has pure codimension $r$ in $Y$ and has rational singularities. As a consequence, we deduce that for a configuration hypersurface with a connected configuration matroid, the corresponding configuration incidence variety has rational singularities. |
| title | On singularities of determinantal hypersurfaces |
| topic | Algebraic Geometry 14B05, 14J17, 32S25 |
| url | https://arxiv.org/abs/2601.22072 |