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1. On the Biplanarity of Blowups

   https://doi.org/10.7155/jgaa.v28i2.2989  

   Eppstein, David (May 2024, Journal of Graph Algorithms and Applications)

   The 2-blowup of a graph is obtained by replacing each vertex with two non-adjacent copies; a graph is biplanar if it is the union of two planar graphs. We disprove a conjecture of Gethner that 2-blowups of planar graphs are biplanar: iterated Kleetopes are counterexamples. Additionally, we construct biplanar drawings of 2-blowups of planar graphs whose duals have two-path induced path partitions, and drawings with split thickness two of 2-blowups of 3-chromatic planar graphs, and of graphs that can be decomposed into a Hamiltonian path and a dual Hamiltonian path. 

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2. An Aubin continuity path for shrinking gradient Kähler–Ricci solitons

   https://doi.org/10.1515/crelle-2024-0053  

   Cifarelli, Charles; Conlon, Ronan J.; Deruelle, Alix (August 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Let 𝐷 be a toric Kähler–Einstein Fano manifold. We show that any toric shrinking gradient Kähler–Ricci soliton on certain toric blowups of C×D satisfies a complex Monge–Ampère equation. We then set up an Aubin continuity path to solve this equation and show that it has a solution at the initial value of the path parameter. This we do by implementing another continuity method. 

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3. Blown-up toric surfaces with non-polyhedral effective cone

   https://doi.org/10.1515/crelle-2023-0022  

   Castravet, Ana-Maria; Laface, Antonio; Tevelev, Jenia; Ugaglia, Luca (May 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We construct projective toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone.As a consequence, we prove that the pseudo-effective cone of the Grothendieck–Knudsen moduli space ${\overline{M}}_{0,n}$\overline{M}_{0,n}of stable rational curves is not polyhedral for $n\ge 10$n\geq 10.These results hold both in characteristic 0 and in characteristic 𝑝, for all primes 𝑝.Many of these toric surfaces are related to an interesting class of arithmetic threefolds that we call arithmetic elliptic pairs of infinite order.Our analysis relies on tools of arithmetic geometry and Galois representations in the spirit of the Lang–Trotter conjecture, producing toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone in characteristic 0 and in characteristic 𝑝, for an infinite set of primes 𝑝 of positive density. 

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4. Projective toric designs, quantum state designs, and mutually unbiased bases

   https://doi.org/10.22331/q-2024-12-03-1546  

   Iosue, Joseph T; Mooney, T C; Ehrenberg, Adam; Gorshkov, Alexey V (December 2024, Quantum)

   Toric $t$-designs, or equivalently $t$-designs on the diagonal subgroup of the unitary group, are sets of points on the torus over which sums reproduce integrals of degree $t$monomials over the full torus. Motivated by the projective structure of quantum mechanics, we develop the notion of $t$-designs on the projective torus, which have a much more restricted structure than their counterparts on full tori. We provide various new constructions of toric and projective toric designs and prove bounds on their size. We draw connections between projective toric designs and a diverse set of mathematical objects, including difference and Sidon sets from the field of additive combinatorics, symmetric, informationally complete positive operator valued measures and complete sets of mutually unbiased bases (MUBs) from quantum information theory, and crystal ball sequences of certain root lattices. Using these connections, we prove bounds on the maximal size of dense $B_{t}modm$sets. We also use projective toric designs to construct families of quantum state designs. In particular, we construct families of (uniformly-weighted) quantum state $2$-designs in dimension $d$of size exactly $d(d+1)$that do not form complete sets of MUBs, thereby disproving a conjecture concerning the relationship between designs and MUBs (Zhu 2015). We then propose a modification of Zhu's conjecture and discuss potential paths towards proving this conjecture. We prove a fundamental distinction between complete sets of MUBs in prime-power dimensions versus in dimension $6$(and, we conjecture, in all non-prime-power dimensions), the distinction relating to group structure of the corresponding projective toric design. Finally, we discuss many open questions about the properties of these projective toric designs and how they relate to other questions in number theory, geometry, and quantum information. 

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5. Adic tropicalizations and cofinality of Gubler models

   https://doi.org/10.1112/blms.70001  

   Foster, Tyler; Payne, Sam (January 2025, Bulletin of the London Mathematical Society)

   Abstract We introduce adic tropicalizations for subschemes of toric varieties as limits of Gubler models associated to polyhedral covers of the ordinary tropicalization. Our main result shows that Huber's adic analytification of a subscheme of a toric variety is naturally isomorphic to the inverse limit of its adic tropicalizations, in the category of locally topologically ringed spaces. The key new technical idea underlying this theorem is cofinality of Gubler models, which we prove for projective schemes and also for more general compact analytic domains in closed subschemes of toric varieties. In addition, we introduce a ‐topology and structure sheaf on ordinary tropicalizations, and show that Berkovich analytifications are limits of ordinary tropicalizations in the category of topologically ringed topoi. 

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