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1. Strongly Compact Cardinals and Ordinal Definability

   https://doi.org/10.1142/S0219061322500106  

   Goldberg, Gabriel (May 2023, Journal of Mathematical Logic)

   This paper explores several topics related to Woodin’s HOD conjecture. We improve the large cardinal hypothesis of Woodin’s HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the first strongly compact cardinal carries an ordinal definable omega Jónsson algebra. We show that if the HOD hypothesis holds and HOD satisfies the Ultrapower Axiom, then every supercompact cardinal is supercompact in HOD. 

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2. On minimal non-σ-scattered linear orders

   https://doi.org/10.1016/j.aim.2024.109540  

   Cummings, James; Eisworth, Todd; Moore, Justin Tatch (April 2024, Advances in Mathematics)

   The purpose of this article is to give new constructions of linear orders which are minimal with respect to being non-σ-scattered. Specifically, we will show that Jensen's diamond principle implies that there is a minimal Countryman line, answering a question of Baumgartner. We also produce the first consistent examples of minimal non-sigma-scattered linear orders of cardinality greater than aleph1, as given a successor cardinal kappa+, we obtain such linear orderings of cardinality kappa+ with the additional property that their square is the union of kappa-many chains. We give two constructions: directly building such examples using forcing, and also deriving their existence from combinatorial principles. The latter approach shows that such minimal non-sigma-scattered linear orders of cardinality kappa+ exist for every infinite cardinal kappa in Gödel's constructible universe, and also (using work of Rinot [28]) that examples must exist at successors of singular strong limit cardinals in the absence of inner models satisfying the existence of a measurable cardinal mu of Mitchell order mu++. 

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3. Schemes Supported on the Singular Locus of a Hyperplane Arrangement in ℙ n

   https://doi.org/10.1093/imrn/rnaa113  

   Migliore, Juan; Nagel, Uwe; Schenck, Henry (May 2020, International Mathematics Research Notices)

   Abstract A hyperplane arrangement in $$\mathbb P^n$$ is free if $R/J$ is Cohen–Macaulay (CM), where $$R = k[x_0,\dots ,x_n]$$ and $$J$$ is the Jacobian ideal. We study the CM-ness of two related unmixed ideals: $$ J^{un}$$, the intersection of height two primary components, and $$\sqrt{J}$$, the radical. Under a mild hypothesis, we show these ideals are CM. Suppose the hypothesis fails. For equidimensional curves in $$\mathbb P^3$$, the Hartshorne–Rao module measures the failure of CM-ness and determines the even liaison class of the curve. We show that for any positive integer $$r$$, there is an arrangement for which $$R/J^{un}$$ (resp. $$R/\sqrt{J}$$) fails to be CM in only one degree, and this failure is by $$r$$. We draw consequences for the even liaison class of $$J^{un}$$ or $$\sqrt{J}$$. 

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4. Galvin’s conjecture and weakly precipitous ideals

   https://doi.org/10.1090/proc/17299  

   Eisworth, Todd (June 2025, Proceedings of the American Mathematical Society)

   We investigate a combinatorial game on ${\mathrm{\omega <\#comment/>}}_1$and show that mild large cardinal assumptions imply that every normal ideal on ${\mathrm{\omega <\#comment/>}}_1$satisfies a weak version of precipitousness. As an application, we show that the Raghavan-Todorčević proof of a longstanding conjecture of Galvin (done assuming the existence of a Woodin cardinal) can be pushed through under much weaker large cardinal assumptions [Forum Math. Pi 8 (2020), p. e15]. 

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5. Sealing from iterability

   Sargsyan, Grigor; Trang, Nam (January 2021, Transactions of the American Mathematical Society)

   null (Ed.)

   We show that if V has a proper class ofWoodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy (as defined in the paper) then Sealing holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to other work by the authors where it is shown that Sealing holds in a generic extension of a certain minimal universe. The current theorem is more general in that no minimality assumption is needed. A corollary of the main theorem is that Sealing is consistent relative to the existence of a Woodin cardinal which is a limit of Woodin cardinals. This improves significantly on the first consistency of Sealing obtained by W.H. Woodin. The Largest Suslin Axiom (LSA) is a determinacy axiom isolated byWoodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let LSA-over-uB be the statement that in all (set) generic extensions there is a model of LSA whose Suslin, co-Suslin sets are the universally Baire sets. The other main result of the paper shows that assuming V has a proper class of inaccessible cardinals which are limit of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, in the universe V [g], where g is V -generic for the collapse of the successor of the least strong cardinal to be countable, the theory LSA-over-UB fails; this implies that LSA-over-UB is not equivalent to Sealing (over the base theory of V [g]). This is interesting and somewhat unexpected, in light of other work by the authors. Compare this result with Steel’s well-known theorem that “AD in L(R) holds in all generic extensions” is equivalent to “the theory of L(R) is sealed” in the presence of a proper class of measurable cardinals. 

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