2018/03/29 by Cristian Anghel, Iustin Coandă, Nicolae Manolache · 2 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Nonlinear Waves and Solitons #Algebraic structures and combinatorial models
paper · doi:10.1090/memo/1209
We provide a complete classification of globally generated vector bundles with first Chern class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c 1 less-than-or-equal-to 5"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo> ≤ </mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">c1 ≤ 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the projective plane and with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c 1 less-than-or-equal-to 4"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo> ≤ </mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">c1 ≤ 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the projective <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -space for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to 3"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n ≥ 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This reproves and extends, in a systematic manner, previous results obtained for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c 1 less-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo> ≤ </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">c1 ≤ 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by Sierra and Ugaglia [J. Pure Appl. Algebra 213(2009), 2141–2146], and for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c 1 equals 3"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">c1 = 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by Anghel and Manolache [Math. Nachr. 286(2013), 1407–1423] and, independently, by Sierra and Ugaglia [J. Pure Appl. Algebra 218(2014), 174–180]. It turns out that the case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c 1 equals 4"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">c1 = 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is much more involved than the previous cases, especially on the projective 3-space. Among the bundles appearing in our classification one can find the Sasakura rank 3 vector bundle on the projective 4-space (conveniently twisted). We also propose a conjecture concerning the classification of globally generated vector bundles with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c 1 less-than-or-equal-to n minus 1"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo> ≤ </mml:mo> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">c1 ≤ n - 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the projective <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -space. We verify the conjecture for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n less-than-or-equal-to 5"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n ≤ 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .