<p>We study induced additive actions on projective hypersurfaces, i.e. effective regular actions of the algebraic group <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1976_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb G_a^m\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">G</mi> <mi>a</mi> <mi>m</mi> </msubsup> </math></EquationSource> </InlineEquation> with an open orbit that can be extended to a regular action on the ambient projective space. It is known that the degree of a hypersurface <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1976_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\subseteq \mathbb {P}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>⊆</mo> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> admitting an induced additive action cannot be greater than <i>n</i> and there is a unique such hypersurface of degree <i>n</i>. We give a complete classification of hypersurfaces <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1976_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\subseteq \mathbb {P}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>⊆</mo> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> admitting an induced additive action of degrees from <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1976_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(n-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1976_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(n-3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Projective Hypersurfaces of High Degree Admitting an Induced Additive Action

  • Ivan Beldiev

摘要

We study induced additive actions on projective hypersurfaces, i.e. effective regular actions of the algebraic group \(\mathbb G_a^m\) G a m with an open orbit that can be extended to a regular action on the ambient projective space. It is known that the degree of a hypersurface \(X\subseteq \mathbb {P}^n\) X P n admitting an induced additive action cannot be greater than n and there is a unique such hypersurface of degree n. We give a complete classification of hypersurfaces \(X\subseteq \mathbb {P}^n\) X P n admitting an induced additive action of degrees from \(n-1\) n - 1 to \(n-3\) n - 3 .