By Mumford.

ISBN-10: 0387112901

ISBN-13: 9780387112909

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We take another disk ∆† := {t ∈ C : |t| < ε} where ε is sufficiently small. Suppose that M is a smooth complex 3-manifold, and Ψ : M → ∆ × ∆† is a proper flat surjective holomorphic map. g. ) We set Mt := Ψ−1 (∆ × {t}) and πt := Ψ|Mt : Mt → ∆ × {t}. Since M is smooth and dim ∆† = 1, the composite map pr2 ◦ Ψ : M → ∆† is a submersion, and so Mt is smooth. We say that Ψ : M → ∆ × ∆† is a deformation family of π : M → ∆ if π0 : M0 → ∆ × {0} coincides with π : M → ∆. By convention, we often denote ∆ × {t} simply by ∆, and we say that πt : Mt → ∆ is a deformation of π : M → ∆.

3 Let N be a line bundle on the projective line P1 such h that N ⊗m ∼ = OP1 (− i=1 mi pi ). Then N ⊗n has a meromorphic section τ which has a pole of order ni (0 ≤ ni ≤ mi ) at pi and is holomorphic on P1 \ {p1 , p2 , . . , ph } if and only if the following inequality holds: n1 + n2 + · · · + nh m1 + m2 + · · · + m h ≥ . 5 Supplement: Example of computation of discriminant loci 53 Proof. 1. + · · · + mh . 3), n1 + n2 + · · · + nh − nr ≥ 0. We separate into two cases according to whether n1 + n2 + · · · + nh − nr is positive (Case 1) or zero (Case 2).

G. ) We set Mt := Ψ−1 (∆ × {t}) and πt := Ψ|Mt : Mt → ∆ × {t}. Since M is smooth and dim ∆† = 1, the composite map pr2 ◦ Ψ : M → ∆† is a submersion, and so Mt is smooth. We say that Ψ : M → ∆ × ∆† is a deformation family of π : M → ∆ if π0 : M0 → ∆ × {0} coincides with π : M → ∆. By convention, we often denote ∆ × {t} simply by ∆, and we say that πt : Mt → ∆ is a deformation of π : M → ∆. We introduce a special class of deformation families of a degeneration. e. any irreducible 1 “Proper” means that all fibers are compact.

### Abelian varieties by Mumford.

by Kenneth

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