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• W1970598762 abstract "In our earlier paper it was proved that the singular locus of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A Subscript g> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>g</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>A_{g}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (coarse moduli space of principally polarized abelian varieties over <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper C> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>C</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) is expressed as the union of irreducible varieties <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A Subscript g Baseline left-parenthesis p comma alpha right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>g</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>A_{g}(p,alpha )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> representing abelian varieties with an order <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=application/x-tex>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> automorphism of fixed entire representation. In this paper we prove that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A Subscript g Baseline left-parenthesis p comma alpha right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>g</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>A_{g}(p,alpha )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an irreducible component of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=Sing upper A Subscript g Baseline> <mml:semantics> <mml:mrow> <mml:mtext>Sing</mml:mtext> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>g</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>text {Sing} A_{g}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if for a general element of this variety its automorphism group modulo <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-brace plus-or-minus 1 right-brace> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mo>±<!-- ± --></mml:mo> <mml:mn>1</mml:mn> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{pm 1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G Subscript plus> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>+</mml:mo> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>G_{+}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, satisfies the equivalent conditions: <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G Subscript plus Baseline equals mathematical left-angle alpha mathematical right-angle> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>+</mml:mo> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mo fence=false stretchy=false>⟨<!-- ⟨ --></mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo fence=false stretchy=false>⟩<!-- ⟩ --></mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>G_{+}=langle alpha rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N Subscript upper G Sub Subscript plus Baseline left-parenthesis mathematical left-angle alpha mathematical right-angle right-parenthesis equals mathematical left-angle alpha mathematical right-angle> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>+</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mo fence=false stretchy=false>⟨<!-- ⟨ --></mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo fence=false stretchy=false>⟩<!-- ⟩ --></mml:mo> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mo fence=false stretchy=false>⟨<!-- ⟨ --></mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo fence=false stretchy=false>⟩<!-- ⟩ --></mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>N_{G_{+}}(langle alpha rangle )=langle alpha rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We illustrate how these results can be used by studying the case <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g equals 4> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>g=4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p equals 5> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>p=5</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W1970598762 created "2016-06-24" @default.
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• W1970598762 date "2011-06-16" @default.
• W1970598762 modified "2023-10-03" @default.
• W1970598762 title "On the irreducible components of the singular locus of 𝐴_{𝑔}. II" @default.
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