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Volumes of divisors in a family and variation of singularities of linear systems.

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Title: Volumes of divisors in a family and variation of singularities of linear systems.
Authors: Jiao, Junpeng1 (AUTHOR)
Source: IMRN: International Mathematics Research Notices. Aug2026, Vol. 2026 Issue 15, p1-19. 19p.
Subject Terms: *Algebraic varieties, *Deformations of singularities, *Mathematical singularities
Abstract: We study the behavior of volumes of divisors in a family. We show that the volume of a divisor on the generic fiber equals the infimum of its volumes on fibers over any dense subset of the base. As an application, we show that the volume function is upper semicontinuous in flat families with reduced and irreducible fibers. Given |$\epsilon>0$|⁠ , we also prove that for a |$\mathbb{Q}$| -Cartier divisor |$B$| on a family of varieties |$X\rightarrow T$|⁠ , if |$(X_{t},|B_{t}|_{\mathbb{Q}})$| is |$\epsilon $| -lc and the volume of |$B_{t}$| is constant for densely many closed points |$t\in T$|⁠ , then the generic fiber |$(X_\eta ,|B_\eta |_{\mathbb{Q}})$| is also |$\epsilon $| -lc. Throughout this paper, the ground field is |$\mathbb{C}$|⁠. By a variety , we mean an integral, separated scheme of finite type over a field (possibly non-algebraically closed). [ABSTRACT FROM AUTHOR]
Database: Academic Search Index
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  Data: Volumes of divisors in a family and variation of singularities of linear systems.
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  Data: <searchLink fieldCode="AR" term="%22Jiao%2C+Junpeng%22">Jiao, Junpeng</searchLink><relatesTo>1</relatesTo> (AUTHOR)
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  Data: <searchLink fieldCode="JN" term="%22IMRN%3A+International+Mathematics+Research+Notices%22">IMRN: International Mathematics Research Notices</searchLink>. Aug2026, Vol. 2026 Issue 15, p1-19. 19p.
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  Data: *<searchLink fieldCode="DE" term="%22Algebraic+varieties%22">Algebraic varieties</searchLink><br />*<searchLink fieldCode="DE" term="%22Deformations+of+singularities%22">Deformations of singularities</searchLink><br />*<searchLink fieldCode="DE" term="%22Mathematical+singularities%22">Mathematical singularities</searchLink>
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  Label: Abstract
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  Data: We study the behavior of volumes of divisors in a family. We show that the volume of a divisor on the generic fiber equals the infimum of its volumes on fibers over any dense subset of the base. As an application, we show that the volume function is upper semicontinuous in flat families with reduced and irreducible fibers. Given |$\epsilon>0$|⁠ , we also prove that for a |$\mathbb{Q}$| -Cartier divisor |$B$| on a family of varieties |$X\rightarrow T$|⁠ , if |$(X_{t},|B_{t}|_{\mathbb{Q}})$| is |$\epsilon $| -lc and the volume of |$B_{t}$| is constant for densely many closed points |$t\in T$|⁠ , then the generic fiber |$(X_\eta ,|B_\eta |_{\mathbb{Q}})$| is also |$\epsilon $| -lc. Throughout this paper, the ground field is |$\mathbb{C}$|⁠. By a variety , we mean an integral, separated scheme of finite type over a field (possibly non-algebraically closed). [ABSTRACT FROM AUTHOR]
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    Identifiers:
      – Type: doi
        Value: 10.1093/imrn/rnag169
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      – Code: eng
        Text: English
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      Pagination:
        PageCount: 19
        StartPage: 1
    Subjects:
      – SubjectFull: Algebraic varieties
        Type: general
      – SubjectFull: Deformations of singularities
        Type: general
      – SubjectFull: Mathematical singularities
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      – TitleFull: Volumes of divisors in a family and variation of singularities of linear systems.
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              M: 08
              Text: Aug2026
              Type: published
              Y: 2026
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              Value: 2026
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