Erratum: Integral points and effective cones of moduli spaces of stable maps Duke Mathematical Journal 120 (2003), no.3, 577-599 Brendan Hassett and Yuri Tschinkel

Abstract We correct a sign error in the published version of the paper.

We are grateful to Ana-Maria Castravet for pointing out a sign error on page 595. The second paragraph should have read:

We extract the inequalities

d_{s+1} − d_{s} ≥ −(n − s − 1)d_{2}.

Adding together the inequalities

d_{n} − d_{n−1} ≥ 0
d_{n−1} − d_{n−2} ≥ −d_{2}
…
d_{4} − d_{3} ≥ −(n − 4)d_{2}
d_{3} ≥ −(n − 4)d_{2}

gives

d_{n} ≥ −\frac{n^{2} − 5n + 4}{2}d_{2}.

Combining with inequality (2), we obtain

n(n − 1)d_{2} ≥ −(n − 4)(n − 1)d_{2},

hence d_{2} ≥ 0.

To complete the proof of Theorem 4.1, we use the curve

W ≃ ℙ^{1} := \overline{\{ρ_{t}(α) : t ∈ 𝔾_{m}\}} ⊂ Y_{α}

introduced in the proof of Lemma 3.2. This is nef relative to Γ and satisfies

W · B[j] = \begin{cases}0 & if j ≠ n, \\ 2 & if j = n, \end{cases}

which implies d_{n} ≥ 0. Combining this with inequality (2) gives d_{j} ≥ 0 for each j. □