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Sections of del Pezzo fibrations over $$\mathbf P^1$$

Sho Tanimoto

Mori’s Bend and Break shows that if we deform a rational curve while fixing two points, then it breaks into the union of rational curves. However, in general it is difficult to control the number of components of a breaking curve and their properties. In this talk I will talk about our recent result for sections of del Pezzo fibrations over $$\mathbf P^1$$ which we call as Movable Bend and Break, i.e., one can break a free section of high height to the union of one free section and one vertical free curve. Then we discuss several applications of this result, Batyrev’s conjecture on the number of components of the space of sections, the irreducibility of the space of sections for certain del Pezzo fibrations, and Batyrev’s heuristic for del Pezzo fibrations. This is joint work with Brian Lehmann.