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<p>We formulate Vojta’s conjecture for smooth weighted projective varieties, weighted multiplier ideal sheaves, and weighted log pairs and prove that all three versions of the conjecture are equivalent. In the process, we introduce generalized weighted general common divisors and express them as heights of weighted projective spaces blown-up relative to an exceptional divisor. Furthermore, we prove that assuming Vojta’s conjecture for weighted projective varieties one can bound the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2024_804_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">\(\log \mathrm{h_{wgcd}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>log</mo> <msub> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">wgcd</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> for any subvariety of codimension <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2024_804_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(\geqslant 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>⩾</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and a finite set of places <i>S</i>. An analogous result is proved for weighted homogeneous polynomials with integer coefficients.</p>

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Vojta’s conjecture on weighted projective varieties

  • 期刊论文

  • 发表日期：

    2025年1月27日

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  • Sajad Salami,
  • Tony Shaska

摘要

We formulate Vojta’s conjecture for smooth weighted projective varieties, weighted multiplier ideal sheaves, and weighted log pairs and prove that all three versions of the conjecture are equivalent. In the process, we introduce generalized weighted general common divisors and express them as heights of weighted projective spaces blown-up relative to an exceptional divisor. Furthermore, we prove that assuming Vojta’s conjecture for weighted projective varieties one can bound the \(\log \mathrm{h_{wgcd}}\) log h wgcd for any subvariety of codimension \(\geqslant 2\) ⩾ 2 and a finite set of places S. An analogous result is proved for weighted homogeneous polynomials with integer coefficients.