Multiplications of maximal rank in the cohomology of ℙ1×ℙ1

• Autores: Salvatore Giuffrida, Renato Maggioni, Riccardo Re
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 63, Fasc. 1, 2012, págs. 1-10
• Idioma: inglés
• DOI: 10.1007/s13348-010-0011-y
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• Resumen

  • Let {Q=\mathbb{P}^1\times\mathbb{P}^1} and let {C\subseteq Q} be a curve of type (a, b) having equation F = 0. The main purpose of this paper is to analize the multiplicative structure of the bi-graded module {H^{1}_{*}\fancyscript{O}_Q} , in particular to prove that for any r, s ≥ 0 the multiplication map {H^{1}\fancyscript{O}_Q(r,-s)\stackrel{F}{\longrightarrow} H^{1}\fancyscript{O}_Q(r+a,-s+b)} induced by F has maximal rank for the general C of type (a, b). Interpretations of this problem in the contexts of multilinear algebra and differential algebra are emphasized.