Algebraic Curves and Riemann Surfaces Errata List Page 52, I think there is a small error. You take a triangulation in the image surface Y and lift it to the base surface X, then you calculate the corresponding Euler number, hence it is understood that your lift is a triangulation of X. However, this is in general not the case with your restricted definition of a triangulation (page 50), namely, you do not allow that the intersection of two triangles contains exactly two vertices. But if you take a triangle side in Y between two branch points, then usually you obtain triangles in X which intersect in exactly two points (on the other hand, you can well allow that triangulations contain such triangles, there is no need for your restricted definition). [Thanks to Paul Schmutz Schaller] Page 129, in the def. of support we need to take the closure of the points where D(p) \not= 0, else you do not get a finite support on a compact surface; [Thanks to Angelo Lopez] In the proof of lemma 1.15 I think we need to take M = min {ord_{p_{i}} g_{j}}, else the proof does not work. [Thanks to Angelo Lopez] P141, the 2\pi i is missing in the calculation of the integral by residues. [Thanks to Sheldon Katz] page 147, middle of the page Take the vector space P(L(D))............ should probably read Take the projectivisation P(L(D)) of the vector space L(D)...... [Thanks to Bill Schulz] page 148, proof of Proposition 3.8: should be div(h)-D_1 = -D_2 (not D_2) [Thanks to Zhou Changwei, 4/13/2010 email] page 167, Problem V.4.H: one needs that p is not a base point of |D|. [Isaac Goldbring] Page 190 where the first line of the proof of the Surjectivity of RES reads Fix a divisor D on X and a NONlinear functional \phi... ^^^ which is not right. [Thanks to Bill Schulz and Angelo Lopez] Page 204, sentence before the new section starts since is it the canonical map after all. [Thanks to Bill Schulz] Page 245, in part 2 of exercise S, we are asked to show that if P is a Weierstrass point on a nonhyperelliptic curve of genus \geq 3, then n_i \leq 2i-2 for all i \geq 2. The same exercise appears in Griffiths and Harris p.275. But for the Fermat quartic in P^2, which has genus 3 and is nonhyperelliptic, there are Weierstrass points with gap sequence 1,2,5. Cf. Arbarello, Cornalba, Griffiths, and Harris Ex. E-2 and E-11 p. 41-44. Solution: For a non-hyperelliptic curve X, Clifford's theorem implies that h0(kp) < 1+k/2 for 1<=k<=2g-2, UNLESS k=2g-2 and (2g-2)p is a canonical divisor; then you have equality (because h^(K) = g, which is 1+(2g-2)/2). With the definition of a gap number as a number j such that h0((j-1)p) = h0(jp), it is clear that the number of gaps less than or equal to k is k+1-h0(kp). Therefore, applying this to k=2i-2, we see that the number of gaps <= 2i-2 is 2i-1-h0((2i-2)p). By Clifford, this is > 2i-1-i = i-1, if we are not in the exceptional case. Therefore the number of gaps <= 2i-1 is strictly greater than i-1, so is at least i; hence n_i <= 2i-2 as claimed in the statement of that exercise in the book. When does this not work? It works UNLESS K=(2g-2)p. This gives an equality in the Clifford statement, and so we see that in this case n_g <=2g-1 instead. This is exactly the example with the Fermat curve, which you correctly point out has g=3, and n_1=1, n_2=2, and n_3=5. [Thanks to Dave Swinarski] Page 250, proof of lemma 2.1, last line it says The element j \in Jac(x) is independent of p; I believe it should say The element j \in Jac(x) is independent of the choice of \gamma; [Thanks to Bill Schulz] Page 254, Exercise D: As stated, this is incorrect. One should either restrict to divisors of degree zero (and use the restricted map A_0) or take the intersection of A^{-1}(A(D)) with the divisors of degree equal to deg(D). [Thanks to Isaac Goldbring] Lemma 4.2 (p303) where we construct a binary-valued partition of unity. Take X to be (0,infinity) with open cover consisting of the open intervals (0,i) (i>0). The catch is that each (0,i) is the union of all the earlier intervals (0,j) (j