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Is there a good errata for Shafarevich's Basic Algebraic Geometry? I don't seem to be able to find one through google.

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Don't know of any (is the error rate so high that it's not better to simply read the book and deal with errors as one finds them?), but here's an interesting error: in the proof of existence of enough translation-invariant 1-forms on a group variety $G$ over an alg. closed field, the argument implicitly uses the false fact that the Zariski topology on $G \times G$ is the product topology. Look in "Neron Models" for an elegant rather different (and correct) method of proof of this result (which applies more widely too). – BCnrd Nov 10 '10 at 15:50
In I.6.3, the corollary to theorem 7 is false. The correct version is either stated on the source or restricted to proper maps. This error is used in II.1.4 where the set of singular points is argued to be closed. That can be fixed by using the projective closure of the tangent spaces to get a proper projection map. In IV.1.1. the formula for intersection number seems wrong. In II.5.3 the proof of normalization of curves seems lacunary. In III.2.2, lemma, the point x must be smooth. spite of missing hypotheses, arguments, this book is excellent and the proofs can be fixed. – roy smith Nov 10 '10 at 16:31
This kind of question is reasonable and comes up periodically on MO, but ideally there would be a more dedicated place to store information on errata for books. As it is, this discussion will very soon be lost in the vast archives out of sight. Apart from that, it's always wise to specify the edition or printing in question. Publishers used to do multiple printings, often with corrections added, as well as new editions. (Not so common with today's technology.) Books in translation pose special problems: serious misprints affecting the mathematics tend to get introduced. – Jim Humphreys Nov 10 '10 at 20:03
Are any of these errors fixed in the recent 3rd edition? – Marius Kempe Jan 18 '14 at 21:22

Springer printed an Errata supposedly to be included in a study edition of the text to be published in 1977. I purchased the original edition directly from Springer and they mailed me the Errata at a later date. I don't really know what transpired in the intervening years.

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I have the 1974 edition and the 12 page errata booklet. They were mostly typos, which do seem to be fixed in the 1994 (2nd edition of the 1977 version), and not mathematical errors. The mathematical errors mentioned above persist through 1994. – roy smith Nov 10 '10 at 17:18

I don't know if it is included in the errata or not: on page 10, about the proof of the representation, he noted that the proof that there are only finitely many points for which $p(x_0,y_0)/q(x_0,y_0)$ equals the common roots of the denominator of $\phi(t)$ and $\psi(t)$ uses "similar reasons", as noted: "for which this fails, for similar reasons", referring to $q$ and $f$ being coprime. But I think this is due to that fact that the function $p/q$ would become constant on $k(X)$.

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