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Curves in Grassmannians and Spanned Stable Bundles
Author(s) -
Ballico Edoardo
Publication year - 2000
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200012)220:1<5::aid-mana5>3.0.co;2-q
Subject(s) - mathematics , surjective function , vector bundle , grassmannian , combinatorics , linear subspace , rank (graph theory) , embedding , genus , geometry , botany , artificial intelligence , biology , computer science
Fix integers g , r , d with g ≥ 2, r ≥ 2 and d > rg . Let X be a smooth curve of genus g and E a general stable vector bundle on X with rank( E = r and deg( E = d . Assume E spanned and let h E : X → G ( r , H 0 ( X , E ) be the induced map into a Grassmannian. Here we give lower bounds on d which assure that the natural map a E : Λ r ( H 0 ( X , E )) → H 0 ( X , det( E )) is surjective and similar results for general subspaces, V , of H 0 ( X , E ). Use the Plücker embedding to see G ( r , H 0 ( X , E )) as a subvariet of P (Λ r ( H 0 ( X , E ))). According to M. Teixidor i Bigas this type of results are related to the geometry of the curve h E ( X ) ⊂ P (Λ r ( H 0 ( X , E ))) and give the dimension of the linear span of h E ( X ).