Unexpected curves and Togliatti‐type surfaces | Zendy

Szpond Justyna | Zendy

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Unexpected curves and Togliatti‐type surfaces

Author(s) -

Szpond Justyna

Publication year - 2020

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/mana.201800455

Subject(s) - osculating circle , mathematics , degree (music) , dimension (graph theory) , plane curve , pure mathematics , perspective (graphical) , surface (topology) , type (biology) , order (exchange) , plane (geometry) , point (geometry) , ruled surface , mathematical analysis , geometry , ecology , physics , finance , acoustics , economics , biology

The purpose of this note is to establish a direct link between the theory of unexpected hypersurfaces and varieties with defective osculating behavior. We identify unexpected plane curves of degree 4 as sections of a rational surface X B of degree 7 in P 5 with its osculating spaces of order 2 which in every point of X B have dimension lower than expected. We put this result in perspective with earlier examples of surfaces with defective osculating spaces due to Shifrin and Togliatti. Our considerations are rendered by an analysis of Lefschetz Properties of ideals associated with the studied surfaces.

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