We should have included in [Reference Landesman, Vakil and WoodLVW24, Theorem 3.2] the additional hypothesis that the fiber over each point $y \in Y$ is arithmetically Gorenstein. Specifically, the sentence ‘We further require that $\rho^{-1}(y) \subset \pi^{-1}(y) \simeq{\mathbb{P}}^{d-2}_{\kappa(y)}$ is a nondegenerate subscheme for each point $y \in Y$ ’ in [Reference Landesman, Vakil and WoodLVW24, Theorem 3.2] should be changed to ‘We further require that $\rho^{-1}(y) \subset \pi^{-1}(y) \simeq{\mathbb{P}}^{d-2}_{\kappa(y)}$ is a nondegenerate and arithmetically Gorenstein subscheme for each point $y \in Y$ ’.

The error in our proof duplicated the same error in the proof and statement of [Reference Casnati and EkedahlCE96, Theorem 2.1], where it is actually implicitly assumed that each fiber is arithmetically Gorenstein. To understand in more detail where the error is coming from, we can restrict to the case that Y is itself the spectrum of a field. Then, in [Reference Casnati and EkedahlCE96, Step A, p. 443], they show that $X \to Y$ has an embedding $i: X \to \mathbb P$ for $\mathbb P$ a projective bundle over Y, with i arithmetically Gorenstein and nondegenerate. If one starts merely with a nondegenerate embedding $i': X \to \mathbb P$ , it may fail to be arithmetically Gorenstein. However, once one assumes $i': X \to \mathbb P$ is arithmetically Gorenstein and nondegenerate, they show any such embedding i ^′ is equivalent to the above embedding $i: X \to \mathbb P$ . Therefore, once one includes this hypothesis that each fiber is arithmetically Gorenstein, one may then continue to run the proof as we described.

For a concrete example where the statement of [Reference Landesman, Vakil and WoodLVW24, Theorem 3.2] is wrong without this assumption, consider the case that X has degree $d = 4$ over Y, $Y = \textrm{Spec} \mathbb C$ , and $X \subset\mathbb P^2$ is the disjoint union of three points on a line and one point off the line. Then X is not a complete intersection of two quadrics (since any quadric containing X must contain the line). Since X has codimension 2 and is not a complete intersection, it is not arithmetically Gorenstein by [Reference MiglioreMig98, Example 4.1.11(c)], even though X is nondegenerate.

Interestingly, in [Reference Casnati and NotariCN07, Theorem 2.2], they did include the hypothesis that each fiber is arithmetically Gorenstein as well as nondegenerate, even though they only noted the importance of mentioning that the fibers are nondegenerate. As we were following their article, we correctly included the arithmetically Gorenstein hypothesis in our statement of [Reference Landesman, Vakil and WoodLVW24, Corollary 3.5].

The above change necessitates a few additional minor changes.

First, [Reference Landesman, Vakil and WoodLVW24, Theorem 3.2(iii)] states that the fibers $\rho^{-1}(y) \subset\pi^{-1}(y)$ are arithmetically Gorenstein, but this should be removed from that part, as it is now part of the hypotheses.

We should add a point (10) to [Reference Landesman, Vakil and WoodLVW24, Remark 3.3] so as to mention that we must assume the fibers $\rho^{-1}(y) \subset \pi^{-1}(y)$ are arithmetically Gorenstein.

Also, in the proof of [Reference Landesman, Vakil and WoodLVW24, Lemma 3.15], we can delete the sentence ‘By Theorem 3.2(iii), $X \to \mathbb P\mathscr E$ is an arithmetically Gorenstein subscheme’ and we can also delete the word ‘Gorenstein’ in the following sentence. The remainder of the proof goes through as written, as this sentence and word were not needed. The point is that one can deduce that [Reference Landesman, Vakil and WoodLVW24, (3.16)] is a minimal free resolution from [Reference Buchsbaum and EisenbudBE77, Theorem 2.1(1)] and its proof, which does not make any assumption that $R/J$ is Gorenstein, as is explained in the final paragraph of [Reference Landesman, Vakil and WoodLVW24, Lemma 3.15].

We thank Enrico Schlesinger for pointing out this oversight to us.