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I'm trying to learn a little about the Lefschetz properties and to start off have been reading Migliore and Nagel's survey article: https://arxiv.org/abs/1109.5718. I'm new to the area, and I have a few questions on terminology that I haven't been able to find by looking up (probably it's out there, but I haven't found it). Hopefully someone here can help.

  1. On page 333 (page 7 of the pdf), the authors mention a "general complete intersection with fixed generator degrees". I am not sure what is meant by this, and how it compares to an arbitrary artinian complete intersection. For that matter, I would also like to know how it follows from Theorem 1.1 and semi-continuity that such a complete intersection has the WLP and SLP. What is meant here by semi-continuity?
  2. On page 338 (page 10 of the pdf), they mention a "general artinian reduction" of a Gorenstein ring. I don't know what this is! I don't know a lot about Gorenstein rings beyond the definition and have never heard of an artinian reduction before. Can anyone explain it to me or point me in the right direction?

Thanks in advance!

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    $\begingroup$ "General" in algebraic geometry, and related areas, means that the construction or statement is valid for an open dense set of the parameter space. Often it is left up to you to decide what the parameter space is, e.g. in the case of complete intersection it might be the space of $n$-tuples of defining polynomials. Perhaps you can ask the authors for more specific questions about their paper. $\endgroup$
    – Donu Arapura
    Commented Mar 3, 2023 at 1:48
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    $\begingroup$ General artinian reduction usually means you quotient by the appropriate number of generic linear forms to produce an artinian algebra. $\endgroup$
    – Sam Hopkins
    Commented Mar 3, 2023 at 14:17
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    $\begingroup$ In fact on pg. 6 (not pg. 10) of the pdf you linked to, it says exactly that: "If $A$ is a Gorenstein ring of dimension $d$, then $A$ is said to have the WLP if a general artinian reduction of $A$ has the WLP, that is, if $A/\langle L_1, \ldots, L_d \rangle$ has the WLP, where $L_1,\ldots,L_d \in A$ are general forms of degree 1." $\endgroup$
    – Sam Hopkins
    Commented Mar 3, 2023 at 16:13

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