Line Configurations in ℙ³ 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 \(\mathbb{P}^3\) with homogeneous ideal \(I\), let \(\alpha(I)\) 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 \(\alpha(I^{(2)})\ge\alpha(I)+1\). Janssen proved (J. Pure Appl. Algebra 219 (2015)) that an arithmetically Cohen–Macaulay configuration with \(\alpha(I^{(2)})=\alpha(I)+1\) is contained in a plane or is a pseudostar, that is, the set of all pairwise intersections of \(m\ge3\) 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 \(\alpha(I^{(2)})=\alpha(I)+1\) 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 \(\alpha(I)+1\) 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 \(\alpha(I)\le2\) 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.