Seminar Algebra and Geometry: Jérémy Blanc (Universität Basel)

When is the blow-up of points or curves in the projective space a weak Fano threefold?

A classical theorem says that del Pezzo surfaces are, with the exception of the smooth quadric, blow-ups of the plane into at most 8 points, such that no three are collinear, no 6 on the same conic and no 8 on the same cubic, singular at one of the 8 points. Moreover, a general set of at most 8 points of the plane satisfies these conditions.

I will describe the theorem, give the proof, and describe the generalisation to dimension 3, by considering blow-ups of points and curves in the projective space. We can get similar descriptions, the conditions of generality are now in terms of multisecant lines, conic or twisted cubics.

Joint work with Stéphane Lamy.

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