<p>A discriminantal hyperplane arrangement <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {B}(n,k,\mathcal {A})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is constructed from a given generic hyperplane arrangement <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>. The arrangement <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> is classified as either very generic or non-very generic according to the combinatorial structure of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {B}(n,k,\mathcal {A})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In particular, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> is regarded as non-very generic if the intersection lattice of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {B}(n,k,\mathcal {A})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> contains at least one non-very generic intersection, namely, an intersection that does not satisfy a specific rank condition established by Athanasiadis. In this paper, we present arithmetic criteria characterizing non-very generic intersections in discriminantal arrangements, and we complete and correct a previous result of Libgober and the third author concerning rank 2 intersections in such arrangements.</p>

Arithmetic non-very generic arrangements

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2026年3月19日

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Pragnya Das,
Takuya Saito,
Simona Settepanella

摘要

A discriminantal hyperplane arrangement $\mathcal {B}(n,k,\mathcal {A})$ $B (n, k, A)$ is constructed from a given generic hyperplane arrangement $\mathcal {A}$ $A$ . The arrangement $\mathcal {A}$ $A$ is classified as either very generic or non-very generic according to the combinatorial structure of $\mathcal {B}(n,k,\mathcal {A})$ $B (n, k, A)$ . In particular, $\mathcal {A}$ $A$ is regarded as non-very generic if the intersection lattice of $\mathcal {B}(n,k,\mathcal {A})$ $B (n, k, A)$ contains at least one non-very generic intersection, namely, an intersection that does not satisfy a specific rank condition established by Athanasiadis. In this paper, we present arithmetic criteria characterizing non-very generic intersections in discriminantal arrangements, and we complete and correct a previous result of Libgober and the third author concerning rank 2 intersections in such arrangements.