Warning: The pdf version contains mistakes and typos that have been corrected in the printed version. Here, only those in the printed version are listed.

Typos:
1. p54,l-9: The notation is not very clear at this point: The Galois groups acts on the etale cohomolo\ gy of the base change of $X$ to an algebraic closure $\bar X=X\times_k\bar k$.

2. p29. Example 3.11: The map given to describe the elliptic fibration is to be understood as a rational map. On the other chart, the map is given by the other factors of the two sums. Also, the first component should be $x_0^2+\xi^2x_1^2$. (Thanks to Oliver Roendigs.)

3. p75, line before Remark 4.5: It should read b_2(Y_0\times\bar k)>3$4. p81, footnote: The quotient should by$Pic(T')$(and not$Pic(T)$) 5. p82, l11: with the scheme$M_d$6. p82, l21:$\pi\colon M_d\to N$7. p219, l-3: The reference should be [236, III.Prop.9.7] 8. p220,l14: In equation for Fermat quartic also x_0 should be cubed. 9. p314: There is a sign mistake in the definition of the ring structure. The last summand should be -(-\lambda\mu'+\lambda'\mu+(x.y))f. (Thanks to Emma Brakkee for pointing this out to me.) 10. p323,l-6: quartic(!) surface 11. p336, Corollary 1.14:$\varphi(m)$is of course not the order of$\mu_m$, but the order of the Galois group of the corresponding cyclotomic field. 12. p405,l-10: At this point one should recall the notation from Section 4.4.1 that$\bar X=X\times_k\bar k$. 13. p419,Cor. 1.13, Cor. 1.15: \hat Br_X has the wrong font. 14. ~ Math: 1. p.118, Theorem 4.1: This is not correctly stated. Borel's theorem indeed needs the assumption of the group being torsion free (and so the quotient being smooth). The argument how to reduce to this case is however correct. (Thanks to Ariyan Javanpeykar for pointing it out and explanation concerning this point.) 2. p246, Proposition 5.4: This is wrong as stated. One has to assume that$X\to {\mathbb P}^1$has no multiple fibre. In fact, in the second approach this was explicitly used. How exactly this enters the first approach mentioned there still eludes me. Correspondingly, (i) in Corollary 5.5 is just wrong. The map from the Tate--Safarevic group to the Weil--Chatelet group is injective, but its cokernel is$\bigoplus H_1(X_t,{\mathbb Q}/{\mathbb Z})$. See Friedman's book [185] for a discussion. (Thanks to Igor Dolgachev for pointing this out to me.) 3. p343,l-8:$f$cannot be symplectic, just apply the Lefschetz fixed point formula. (Thanks to Serge Cantat for pointing this out.) 4. p343, Section 2.5: As Serge Cantat pointed out to me, K3 surfaces that are double cover of Enriques surfaces have large automorphism group. Indeed, the automorphism group of an Enriques surface is a subgroup of finite index of the orthogonal group of its integral cohomology. 5. p384: Arend Bayer explained to me how to prove that$E^\perp$for$E\$ spherical on a K3 surface of Picard number one is not empty.