Erratum: Symmetries of Fano varieties (Journal fur die Reine und Angewandte Mathematik (2025) 819 (89–133) DOI: 10.1515/crelle-2024-0077)

We state a correction of an error in the n D 4 case of Theorem 3 of the paper [L. Esser, L. Ji and J. Moraga, Symmetries of Fano varieties, J. Reine Angew. Math. 819 (2025), 89–133]. The classification of maximal examples for n D 4 in [1, Theorem 3] is incorrect. In the original article, we state that if the symmetric group S_k acts faithfully on a complete simplicial toric variety X of dimension 4, then k 6, and moreover that if k D 6, then X Š P⁴ or X Š P² P². The bound k 6 is correct but the classification of optimal examples misses several cases. The correct classification is as follows: Theorem 1 ([2, Theorem 3.2]). Suppose that the group S₆ acts faithfully on a complete simplicial toric variety X of dimension 4. Then X is isomorphic to one of the following: (1) P⁴, (2) P² P², (3) P_P3.O O.a// for some even integer a, or (4) P.1; 1; 1; 1; a/ for some positive even integer a. The error in the proof of [1, Theorem 3] occurs on page 105, when considering the case that S_k Aut⁰.X/. The argument refers to [1, Lemma 4.4], which states that S_k Aut⁰.X/ implies that the dimension n of X satisfies n 3 if k D 5;6 and n k 2 if k 7. Furthermore, when these inequalities are equalities, X Š Pⁿ. This lemma implies that if n D 4, then k 6. However, equality is never attained for any value of k when n D 4; hence the conclusion that S₆ Aut⁰.X/ implies X Š P⁴ is incorrect. The full classification and proof of Theorem 1 appears in recent work of Naito [2, Section 3]. The methods are similar to those in [1, Section 4], using Cox’s description of Aut⁰.X/ in terms of the fan of X to constrain the possible arrangements of rays.