Equations and syzygies of the first secant variety to a smooth curve

2011/12/14 by Peter Vermeire
Mathematics · Engineering · #Tensor decomposition and applications #Algebraic Geometry and Number Theory #Advanced Numerical Analysis Techniques

paper · pdf · doi:10.1090/s0002-9939-2011-11392-3

Abstract

We show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a linearly normal smooth curve embedded by a line bundle of degree at least <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 g plus 3 plus p"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>g</mml:mi> <mml:mo>+</mml:mo> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2g+3+p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then the secant variety to the curve satisfies <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N Subscript 3 comma p"> <mml:semantics> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">N_3,p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

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