Kustin–Miller unprojection with complexes

Abstract

A main ingredient for the Kustin–Miller unprojection is the module <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H o m Subscript upper R Baseline left-parenthesis upper I comma omega Subscript upper R Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>Hom</mml:mi> <mml:mi>R</mml:mi> </mml:msub> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>I</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi> ω </mml:mi> <mml:mi>R</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Hom}_R(I, \omega _R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a local Gorenstein ring and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a codimension one ideal with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R slash upper I"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>I</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">R/I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Gorenstein. We prove a method of calculating it in a relative setting using resolutions. We give three applications. In the first we generalise a result of Catanese, Franciosi, Hulek, and Reid ( <italic>Embeddings of curves and surfaces</italic> , Nagoya Math. J. <bold>154</bold> (1999), 185–220). The second and the third are about Tom and Jerry, two families of Gorenstein codimension four rings with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="9 times 16"> <mml:semantics> <mml:mrow> <mml:mn>9</mml:mn> <mml:mo> × </mml:mo> <mml:mn>16</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">9 \times 16</mml:annotation> </mml:semantics> </mml:math> </inline-formula> resolutions.