The Gaussian-Wahl map for trigonal curves

Abstract

If a curve <italic>C</italic> is embedded in projective space by a very ample line bundle <italic>L</italic> , the Gaussian map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi Subscript upper C comma upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal"> Φ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>C</mml:mi> <mml:mo>,</mml:mo> <mml:mi>L</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Phi _{C,L}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is defined as the pull-back of hyperplane sections of the classical Gauss map composed with the Plücker embedding. When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L equals upper K"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>=</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">L = K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the canonical divisor of the curve <italic>C</italic> , the map is known as the Gaussian-Wahl map for <italic>C</italic> . We determine the corank of the Gaussian-Wahl map to be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g plus 5"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>+</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">g + 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all trigonal curves (i.e., curves which admit a 3-to-1 mapping onto the projective line) by examining the way in which a trigonal curve is naturally embedded in a rational normal scroll.