For each <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$k \ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> we give a counterexample to a conjecture of Movasati on the dimension of certain Hodge loci of cubic hypersurfaces in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textbf{P}^{2k+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="bold">P</mi> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> containing two k-planes intersecting in dimension <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$k-3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>-</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. We give similar examples for Hodge loci of cubic hypersurfaces in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textbf{P}^{2k+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="bold">P</mi> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> containing two k-planes intersecting in dimension <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$k-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and for quartic hypersurfaces in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textbf{P}^{2k+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="bold">P</mi> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> containing two k-planes intersecting in dimension <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$k-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Moreover, we present new evidence for Movasati’s conjecture for the values of k for which our type of counterexamples cannot exist, i.e., for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$k=3,4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>.

On a conjecture on Hodge loci of linear combinations of linear subvarieties

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2025年8月20日

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Remke Kloosterman

摘要

For each $k \ge 5$ $k \geq 5$ we give a counterexample to a conjecture of Movasati on the dimension of certain Hodge loci of cubic hypersurfaces in $\textbf{P}^{2k+1}$ $P^{2 k + 1}$ containing two k-planes intersecting in dimension $k-3$ $k - 3$ . We give similar examples for Hodge loci of cubic hypersurfaces in $\textbf{P}^{2k+1}$ $P^{2 k + 1}$ containing two k-planes intersecting in dimension $k-2$ $k - 2$ and for quartic hypersurfaces in $\textbf{P}^{2k+1}$ $P^{2 k + 1}$ containing two k-planes intersecting in dimension $k-2$ $k - 2$ . Moreover, we present new evidence for Movasati’s conjecture for the values of k for which our type of counterexamples cannot exist, i.e., for $k=3,4$ $k = 3, 4$ .