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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. A classification of infinite staircases for Hirzebruch surfaces

   https://doi.org/10.1112/topo.70017  

   Magill, Nicki; Pires, Ana_Rita; Weiler, Morgan (March 2025, Journal of Topology)

   Abstract The ellipsoid embedding function of a symplectic manifold gives the smallest amount by which the symplectic form must be scaled in order for a standard ellipsoid of the given eccentricity to embed symplectically into the manifold. It was first computed for the standard four‐ball (or equivalently, the complex projective plane) by McDuff and Schlenk, and found to contain the unexpected structure of an “infinite staircase,” that is, an infinite sequence of nonsmooth points arranged in a piecewise linear stair‐step pattern. Later work of Usher and Cristofaro‐Gardiner–Holm–Mandini–Pires suggested that while four‐dimensional symplectic toric manifolds with infinite staircases are plentiful, they are highly nongeneric. This paper concludes the systematic study of one‐point blowups of the complex projective plane, building on previous work of Bertozzi‐Holm‐Maw‐McDuff‐Mwakyoma‐Pires‐Weiler, Magill‐McDuff, Magill‐McDuff‐Weiler, and Magill on these Hirzebruch surfaces. We prove a conjecture of Cristofaro‐Gardiner–Holm–Mandini–Pires for this family: that if the blowup is of rational weight and the embedding function has an infinite staircase then that weight must be . We show also that the function for this manifold does not have a descending staircase. Furthermore, we give a sufficient and necessary condition for the existence of an infinite staircase in this family which boils down to solving a quadratic equation and computing the function at one specific value. Many of our intermediate results also apply to the case of the polydisk (or equivalently, the symplectic product of two spheres). 

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3. 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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4. 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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5. 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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