Given a smooth hypersurface X⊂P^{n+1} of degree d⩾2, we study the cones V^h_p⊂P^{n+1} swept out by lines having contact order h⩾2 at a point p∈X. In particular, we prove that if X is general, then for any p∈X and 2⩽h⩽min{n+1,d}, the cone V^h_p has dimension exactly n+2−h. Moreover, when X is a very general hypersurface of degree d⩾2n+2, we describe the relation between the cones V^h_p and the degree of irrationality of k-dimensional subvarieties of X passing through a general point of X. As an application, we give some bounds on the least degree of irrationality of k-dimensional subvarieties of X passing through a general point of X, and we prove that the connecting gonality of X satisfies suitable inequalities
Bastianelli, F., Ciliberto, C., Flamini, F., Supino, P. (2021). Cones of lines having high contact with general hypersurfaces and applications. MATHEMATISCHE NACHRICHTEN.
Cones of lines having high contact with general hypersurfaces and applications
Ciliberto, Ciro;Flamini, Flaminio;
2021-06-05
Abstract
Given a smooth hypersurface X⊂P^{n+1} of degree d⩾2, we study the cones V^h_p⊂P^{n+1} swept out by lines having contact order h⩾2 at a point p∈X. In particular, we prove that if X is general, then for any p∈X and 2⩽h⩽min{n+1,d}, the cone V^h_p has dimension exactly n+2−h. Moreover, when X is a very general hypersurface of degree d⩾2n+2, we describe the relation between the cones V^h_p and the degree of irrationality of k-dimensional subvarieties of X passing through a general point of X. As an application, we give some bounds on the least degree of irrationality of k-dimensional subvarieties of X passing through a general point of X, and we prove that the connecting gonality of X satisfies suitable inequalitiesFile | Dimensione | Formato | |
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