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1. Combinatorics and Real Lifts of Bitangents to Tropical Quartic Curves

   https://doi.org/10.1007/s00454-022-00445-1  

   Cueto, Maria Angelica; Markwig, Hannah (April 2023, Discrete & Computational Geometry)

   Abstract Smooth algebraic plane quartics over algebraically closed fields of characteristic different than two have 28 bitangent lines. Their tropical counterparts often have infinitely many bitangents. They are grouped into seven equivalence classes, one for each linear system associated to an effective tropical theta characteristic on the tropical quartic. We show such classes determine tropically convex sets and provide a complete combinatorial classification of such objects into 41 types (up to symmetry). The occurrence of a given class is determined by both the combinatorial type and the metric structure of the input tropical plane quartic. We use this result to provide explicit sign-rules to obtain real lifts for each tropical bitangent class, and confirm that each one has either zero or exactly four real lifts, as previously conjectured by Len and the second author. Furthermore, such real lifts are always totally-real. 

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2. Rationality of Real Conic Bundles With Quartic Discriminant Curve

   https://doi.org/10.1093/imrn/rnad003  

   Ji, Lena; Ji, Mattie (February 2023, International Mathematics Research Notices)

   We study real double covers of $$\mathbb P^{1}\times \mathbb P^{2}$$ branched over a $(2,2)$-divisor, which are conic bundles with smooth quartic discriminant curve by the second projection. In each isotopy class of smooth plane quartics, we construct examples where the total space is $$\mathbb R$$-rational. For five of the six isotopy classes, we construct $$\mathbb C$$-rational examples with obstructions to rationality over $$\mathbb R$$, and for the sixth class, we show that the models we consider are all rational. Moreover, for three of the five classes with irrational members, we characterize rationality using the real locus and the intermediate Jacobian torsor obstruction of Hassett–Tschinkel and Benoist–Wittenberg. These double cover models were introduced by Frei, Sankar, Viray, Vogt, and the first author, who determined explicit descriptions for their intermediate Jacobian torsors. 

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3. Enumerative geometry of del Pezzo surfaces

   https://doi.org/10.1090/tran/8817  

   Lin, Yu-Shen (February 2024, Transactions of the American Mathematical Society)

   We prove an equivalence between the superpotential defined via tropical geometry and Lagrangian Floer theory for special Lagrangian torus fibres in del Pezzo surfaces constructed by Collins-Jacob-Lin [Duke Math. J. 170 (2021), pp. 1291–1375]. We also include some explicit calculations for the projective plane, which confirm some folklore conjectures in this case. 

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4. Topological Realizations of Line Arrangements

   https://doi.org/10.1093/imrn/rnx190  

   Ruberman, Daniel; Starkston, Laura (August 2017, International Mathematics Research Notices)

   Abstract A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines. The classic version of this would ask for lines in a projective plane over a field. An important variation allows for pseudolines: embedded circles (isotopic to $$\mathbb R\rm{P}^1$$) in the real projective plane. In this article we investigate whether a configuration is realized by a collection of 2-spheres embedded, in symplectic, smooth, and topological categories, in the complex projective plane. We find obstructions to the existence of topologically locally flat spheres realizing a configuration, and show for instance that the combinatorial configuration corresponding to the projective plane over any finite field is not realized. Such obstructions are used to show that a particular contact structure on certain graph manifolds is not (strongly) symplectically fillable. We also show that a configuration of real pseudolines can be complexified to give a configuration of smooth, indeed symplectically embedded, 2-spheres. 

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5. Regimes of Convective Self-Aggregation in Convection-Permitting Beta-Plane Simulations

   https://doi.org/10.1175/JAS-D-22-0222.1  

   Carstens, Jacob D.; Wing, Allison A. (September 2023, Journal of the Atmospheric Sciences)

   Abstract The spontaneous self-aggregation (SA) of convection in idealized model experiments highlights the importance of interactions between tropical convection and the surrounding environment. The authors have shown that SA fundamentally changes with the background rotation in previousf-plane simulations, in terms of both the resulting forms of organized convection and the relative roles of the physical feedbacks driving them. This study considers the dependence of SA on rotation in one large domain on theβplane, introducing an additional layer of complexity. Simulations are performed with uniform thermal forcing and explicit convection. Focuses include statistical and structural analysis of the convective modes, process-oriented diagnostics of how they develop, and resulting mean states. Two regimes of SA emerge within the first 15 days, separated by a critical zone wherefis analogous to 10°–15° latitude. Organized convection at near-equatorial values offprimarily consists of convectively coupled Kelvin waves. Wind speed–surface enthalpy flux feedbacks are the dominant process driving moisture variability early on, then clear-sky shortwave radiative feedbacks are strongest in wave maintenance. In contrast, at higherf, numerous tropical cyclones develop and coexist, dominated by surface flux and longwave processes. Tropical cyclogenesis is most pronounced at intermediatef(analogous to 25°–40°), but are longer-lived at higherf. The resulting modes of SA at lowfdiffer between theseβ-plane simulations (convectively coupled waves) and priorf-plane simulations (weak tropical cyclones or nonrotating clusters). Otherwise, these results provide further evidence for the changing roles of radiative, surface flux, and advective processes in influencing SA asfchanges, as found in our previous study. Significance StatementIn model simulations, convection often self-organizes due to interactions with its surrounding environment. These interactions are relevant in the real-world organization of rainfall and clouds, and may thus be useful to understand for improved prediction of tropical weather and climate. Previous work using a set of simple model experiments with constant Coriolis force showed that at different latitudes, different processes dominate, and different types of organized convection result. This study verifies that finding using a more complex and realistic model, where the Coriolis force varies within the domain to resemble different latitudes. Specifically, the convection here self-organizes into atmospheric waves (periodic disturbances) at low latitudes, and tropical cyclones at high latitudes. 

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