Logarithmic Quot spaces, boundedness, and K-tropicalizations
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arXiv
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| Natura: | Preprint |
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2025
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| author | Kennedy-Hunt, Patrick Ranganathan, Dhruv |
| author_facet | Kennedy-Hunt, Patrick Ranganathan, Dhruv |
| contents | Logarithmic Hilbert and Quot spaces are generalizations of their traditional versions adapted to study pairs and degenerations. The logarithmic Quot spaces of $(X,D)$ parameterize "algebraically transverse" (logarithmically flat) quotient sheaves on degenerations of $X$. We prove boundedness and deduce properness of logarithmic Quot spaces. The results complete the basic foundations of logarithmic Quot spaces and specialize to work of Li-Wu and Maulik and the second author in special cases. Boundedness relies on two results of independent interest. First, we show that for a simple normal crossing pair $(X, D)$ and a subscheme $Z$, there is a smallest logarithmic space ${X}^\flat$ modifying $X$ such that the strict transform of $Z$ is algebraically transverse. Precisely, given $Z$ in $X$, there is a canonical logarithmic space $X^\flat$ over $X$ with the following universal property - an snc logarithmic blowup $X'\to X$ makes the strict transform of $Z$ algebraically transverse if and only if $X'$ is a modification of $X^\flat$. Parallel results hold for arbitrary coherent sheaves. This proves boundedness for logarithmic quotients with fixed tropicalization. A logarithmic quotient sheaf defines a K-tropicalization, an enhancement of tropicalization that is sensitive to scheme structures. The K-tropicalization has the same relationship to K-theory as traditional tropicalization has to Chow, and is related to Gröbner theory and convex geometry via state and secondary polytopes. Using the K-theory of toric bundles, we derive a balancing condition for K-tropicalizations that imposes strong finiteness properties. The second key result is that K-tropicalizations with fixed numerics are parametrized by a finite-dimensional polyhedral complex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08175 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Logarithmic Quot spaces, boundedness, and K-tropicalizations Kennedy-Hunt, Patrick Ranganathan, Dhruv Algebraic Geometry 14D20, 14A21, 14T90 Logarithmic Hilbert and Quot spaces are generalizations of their traditional versions adapted to study pairs and degenerations. The logarithmic Quot spaces of $(X,D)$ parameterize "algebraically transverse" (logarithmically flat) quotient sheaves on degenerations of $X$. We prove boundedness and deduce properness of logarithmic Quot spaces. The results complete the basic foundations of logarithmic Quot spaces and specialize to work of Li-Wu and Maulik and the second author in special cases. Boundedness relies on two results of independent interest. First, we show that for a simple normal crossing pair $(X, D)$ and a subscheme $Z$, there is a smallest logarithmic space ${X}^\flat$ modifying $X$ such that the strict transform of $Z$ is algebraically transverse. Precisely, given $Z$ in $X$, there is a canonical logarithmic space $X^\flat$ over $X$ with the following universal property - an snc logarithmic blowup $X'\to X$ makes the strict transform of $Z$ algebraically transverse if and only if $X'$ is a modification of $X^\flat$. Parallel results hold for arbitrary coherent sheaves. This proves boundedness for logarithmic quotients with fixed tropicalization. A logarithmic quotient sheaf defines a K-tropicalization, an enhancement of tropicalization that is sensitive to scheme structures. The K-tropicalization has the same relationship to K-theory as traditional tropicalization has to Chow, and is related to Gröbner theory and convex geometry via state and secondary polytopes. Using the K-theory of toric bundles, we derive a balancing condition for K-tropicalizations that imposes strong finiteness properties. The second key result is that K-tropicalizations with fixed numerics are parametrized by a finite-dimensional polyhedral complex. |
| title | Logarithmic Quot spaces, boundedness, and K-tropicalizations |
| topic | Algebraic Geometry 14D20, 14A21, 14T90 |
| url | https://arxiv.org/abs/2508.08175 |