Irregular surfaces on hypersurfaces of degree 4 with non-degenerate isolated singularities.

A rank 5 vector bundle G on HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign=" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true"><msubsup><mi mathvariant="double-struck">P</mi><mi mathvariant="double-struck">Q</mi><mn>4</mn></msubsup></mstyle></mtd></mtr></mtable></math> ht is constructed as the homology of a certain Beilinson monad. By the double point formula, HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign=" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true"><msubsup><mi>K</mi><mi>X</mi><mn>2</mn></msubsup></mstyle></mtd></mtr></mtable></math> ht = - 4. The entries of the third row are given by I i >= 2 I i - HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign=" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true"><msubsup><mi>K</mi><mi>X</mi><mn>2</mn></msubsup></mstyle></mtd></mtr></mtable></math> ht . Substituting this value in the double point formula, we get HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign=" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true"><msubsup><mi>K</mi><mi>X</mi><mn>2</mn></msubsup></mstyle></mtd></mtr></mtable></math> ht = -12. [Extracted from the article]