<p>We study projective hypersurfaces <i>X</i> admitting an induced additive action, i.e., an effective action <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathbb {G}_a^m\times X\rightarrow X}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="double-struck">G</mi> <mi>a</mi> <mi>m</mi> </msubsup> <mo>×</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> of the vector group <InlineEquation ID="IEq2"> <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 an action on the ambient projective space. A criterion for normality of such a hypersurface <i>X</i> is given. Also, we prove that for any projective hypersurface <i>Z</i> there exists a hypersurface <i>X</i> with an induced additive action such that the complement to the open <InlineEquation ID="IEq3"> <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>-orbit in <i>X</i> is a projective cone over <i>Z</i>. We introduce a construction that produces non-degenerate hypersurfaces with induced additive action from Young diagrams and study the properties of the hypersurfaces obtained in this way.</p>

错误:搜索内容不能为空,请输入英文关键词
错误:关键词超出字数限制,请精简
高级检索

On normality of projective hypersurfaces with an additive action

  • Ivan Arzhantsev,
  • Ivan Beldiev,
  • Yulia Zaitseva

摘要

We study projective hypersurfaces X admitting an induced additive action, i.e., an effective action \({\mathbb {G}_a^m\times X\rightarrow X}\) G a m × X X of the vector group \(\mathbb {G}_a^m\) G a m with an open orbit that can be extended to an action on the ambient projective space. A criterion for normality of such a hypersurface X is given. Also, we prove that for any projective hypersurface Z there exists a hypersurface X with an induced additive action such that the complement to the open \(\mathbb {G}_a^m\) G a m -orbit in X is a projective cone over Z. We introduce a construction that produces non-degenerate hypersurfaces with induced additive action from Young diagrams and study the properties of the hypersurfaces obtained in this way.