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Abstract:
Let X be a "general" algebraic surface defined by a homogeneous polynomial of degree d, d at least 4 in complex projective space of dimension 3. The Noether-Lefschetz theorem in geometric form, asserts that any algebraic curve C in X must be of the form $X\cap{S}$, for another surface S in the projective space. The analogue of this theorem to higher dimensions, conjectured by Griffiths and Harris was shown to be false by C.Voisin.
In this talk, I will show that a weaker generalisation of this theorem is true and discuss the main circle of ideas.
This talk will be accessible to graduate students.
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