Line Configurations in P^3 Whose Symbolic Square Has Initial Degree α(I)+1 Are Coplanar or Pseudostars: A Negative Answer to a Question of Janssen

For a finite union L of lines in P^3 with homogeneous ideal I, let α(I) be the least degree of a non-zero form in I, and let I^(2) be the second symbolic power of I, the ideal of the forms that vanish to order at least two along every line of L; one always has α(I^(2)) ≥ α(I) + 1. Janssen proved (J. Pure Appl. Algebra 219 (2015)) that an arithmetically Cohen–Macaulay configuration with α(I^(2)) = α(I) + 1 is contained in a plane or is a pseudostar, that is, the set of all pairwise intersections of m ≥ 3 planes no three of which contain a common line. He asked, there and in Oberwolfach Report 9/2015, whether a configuration of lines that is not arithmetically Cohen–Macaulay can satisfy this equality. We show that it cannot. Over an algebraically closed field of arbitrary characteristic, the equality α(I^(2)) = α(I) + 1 holds if and only if L is contained in a plane or is a pseudostar; both kinds of configurations are arithmetically Cohen–Macaulay, so the hypothesis in Janssen's theorem is not needed. The same holds for reduced curves of pure dimension one: the equality holds exactly for plane curves and for pseudostars of lines. In the proof we factor a form of degree α(I) + 1 that is singular along L; the case of an irreducible form is excluded by the following statement, which we prove for arbitrary singularities and in every characteristic: the curves along which an irreducible surface of degree d is singular lie on a surface of degree d − 2. This statement is classical in spirit (adjoint surfaces). For α(I) ≤ 2 the classification already follows from a theorem of Haghighi and Mosakhani (2021), which is known to us only through its abstract and its zbMATH review. Computations, among them an exhaustive check of all 2^35 − 1 sets of lines of the projective space over the field with two elements, agree with the theorem; they are corroboration only and are not used in the proofs. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-13494-011 (Janssen, On the fattening of lines in P^3, Oberwolfach Reports 12 (2015), Report No. 9/2015, first question on p. 514).