Recent zbMATH articles in MSC 14Thttps://www.zbmath.org/atom/cc/14T2021-06-15T18:09:00+00:00WerkzeugMotivic volumes of fibers of tropicalization.https://www.zbmath.org/1460.140382021-06-15T18:09:00+00:00"Usatine, Jeremy"https://www.zbmath.org/authors/?q=ai:usatine.jeremySummary: Let \(T\) be an algebraic torus over an algebraically closed field, let \(X\) be a smooth closed subvariety of a \(T\)-toric variety such that \(U = X \cap T\) is not empty, and let \(\mathscr{L}(X)\) be the arc scheme of \(X\). We consider a tropicalization map on \(\mathscr{L}(X) \setminus \mathscr{L}(X \setminus U)\), the set of arcs of \(X\) that do not factor through \(X \setminus U\). We show that each fiber of this tropicalization map is a constructible subset of \(\mathscr{L}(X)\) and therefore has a motivic volume. We prove that if \(U\) has a compactification with simple normal crossing boundary, then the generating function for these motivic volumes is rational, and we express this rational function in terms of certain lattice maps constructed in Hacking, Keel, and Tevelev's [10] theory of geometric tropicalization. We explain how this result, in particular, gives a formula for \textit{J. Denef} and \textit{F. Loeser}'s [J. Algebr. Geom. 7, No. 3, 505--537 (1998; Zbl 0943.14010)] motivic zeta function of a polynomial. To further understand this formula, we also determine precisely which lattice maps arise in the construction of geometric tropicalization.Detecting tropical defects of polynomial equations.https://www.zbmath.org/1460.141402021-06-15T18:09:00+00:00"Görlach, Paul"https://www.zbmath.org/authors/?q=ai:gorlach.paul"Ren, Yue"https://www.zbmath.org/authors/?q=ai:ren.yue.1|ren.yue"Sommars, Jeff"https://www.zbmath.org/authors/?q=ai:sommars.jeffSummary: We introduce the notion of tropical defects, certificates that a system of polynomial equations is not a tropical basis, and provide two algorithms for finding them in affine spaces of complementary dimension to the zero set. We use these techniques to solve open problems regarding del Pezzo surfaces of degree 3 and realizability of valuated gaussoids on 4 elements.Cycles, cocycles, and duality on tropical manifolds.https://www.zbmath.org/1460.141412021-06-15T18:09:00+00:00"Gross, Andreas"https://www.zbmath.org/authors/?q=ai:gross.andreas"Shokrieh, Farbod"https://www.zbmath.org/authors/?q=ai:shokrieh.farbodSummary: We prove a Poincaré duality for the Chow rings of smooth fans whose support are tropical linear spaces. As a consequence, we show that cycles and cocycles on tropical manifolds are Poincaré dual to each other. This allows us to define pull-backs of tropical cycles along arbitrary morphisms with smooth target.Enumeration of holomorphic cylinders in log Calabi-Yau surfaces. II: Positivity, integrality and the gluing formula.https://www.zbmath.org/1460.141252021-06-15T18:09:00+00:00"Yu, Tony Yue"https://www.zbmath.org/authors/?q=ai:yu.tony-yueSummary: We prove three fundamental properties of counting holomorphic cylinders in log Calabi-Yau surfaces: positivity, integrality and the gluing formula. Positivity and integrality assert that the numbers of cylinders, defined via virtual techniques, are in fact nonnegative integers. The gluing formula roughly says that cylinders can be glued together to form longer cylinders, and the number of longer cylinders equals the product of the numbers of shorter cylinders. Our approach uses Berkovich geometry, tropical geometry, deformation theory and the ideas in the proof of associativity relations of Gromov-Witten invariants by Maxim Kontsevich [\textit{M. Kontsevich} and \textit{Yu. Manin}, Commun. Math. Phys. 164, No. 3, 525--562 (1994; Zbl 0853.14020)]. These three properties provide evidence for a conjectural relation between counting cylinders and the broken lines of Gross, Hacking and Keel [\textit{M. Gross} et al., Publ. Math., Inst. Hautes Étud. Sci. 122, 65--168 (2015; Zbl 1351.14024)].
For part I see [the author, Math. Ann. 366, No. 3--4, 1649--1675 (2016; Zbl 1375.14186)].Forbidden patterns in tropical plane curves.https://www.zbmath.org/1460.520142021-06-15T18:09:00+00:00"Joswig, Michael"https://www.zbmath.org/authors/?q=ai:joswig.michael"Tewari, Ayush Kumar"https://www.zbmath.org/authors/?q=ai:tewari.ayush-kumarSummary: Tropical curves in \(\mathbb{R}^2\) correspond to metric planar graphs but not all planar graphs arise in this way. We describe several new classes of graphs which cannot occur. For instance, this yields a full combinatorial characterization of the tropically planar graphs of genus at most five.Gonality sequences of graphs.https://www.zbmath.org/1460.141422021-06-15T18:09:00+00:00"Aidun, Ivan"https://www.zbmath.org/authors/?q=ai:aidun.ivan"Dean, Frances"https://www.zbmath.org/authors/?q=ai:dean.frances"Morrison, Ralph"https://www.zbmath.org/authors/?q=ai:morrison.ralph"Yu, Teresa"https://www.zbmath.org/authors/?q=ai:yu.teresa"Yuan, Julie"https://www.zbmath.org/authors/?q=ai:yuan.julieTropical hyperelliptic curves in the plane.https://www.zbmath.org/1460.141472021-06-15T18:09:00+00:00"Morrison, Ralph"https://www.zbmath.org/authors/?q=ai:morrison.ralphSummary: Abstractly, tropical hyperelliptic curves are metric graphs that admit a two-to-one harmonic morphism to a tree. They also appear as embedded tropical curves in the plane arising from triangulations of polygons with all interior lattice points collinear. We prove that hyperelliptic graphs can only arise from such polygons. Along the way, we will prove certain graphs do not embed tropically in the plane due to entirely combinatorial obstructions, regardless of whether their metric is actually hyperelliptic.Linear programs and convex hulls over fields of Puiseux fractions.https://www.zbmath.org/1460.901062021-06-15T18:09:00+00:00"Joswig, Michael"https://www.zbmath.org/authors/?q=ai:joswig.michael"Loho, Georg"https://www.zbmath.org/authors/?q=ai:loho.georg"Lorenz, Benjamin"https://www.zbmath.org/authors/?q=ai:lorenz.benjamin.1"Schröter, Benjamin"https://www.zbmath.org/authors/?q=ai:schroter.benjaminSummary: We describe the implementation of a subfield of the field of formal Puiseux series in polymake. This is employed for solving linear programs and computing convex hulls depending on a real parameter. Moreover, this approach is also useful for computations in tropical geometry.
For the entire collection see [Zbl 1334.68018].Tropical superelliptic curves.https://www.zbmath.org/1460.141442021-06-15T18:09:00+00:00"Brandt, Madeline"https://www.zbmath.org/authors/?q=ai:brandt.madeline"Helminck, Paul Alexander"https://www.zbmath.org/authors/?q=ai:helminck.paul-alexanderSummary: We present an algorithm for computing the Berkovich skeleton of a superelliptic curve \(y_n = f(x)\) over a valued field. After defining superelliptic weighted metric graphs, we show that each one is realizable by an algebraic superelliptic curve when \(n\) is prime. Lastly, we study the locus of superelliptic weighted metric graphs inside the moduli space of tropical curves of genus \(g\).\(K3\) polytopes and their quartic surfaces.https://www.zbmath.org/1460.141432021-06-15T18:09:00+00:00"Balletti, Gabriele"https://www.zbmath.org/authors/?q=ai:balletti.gabriele"Panizzut, Marta"https://www.zbmath.org/authors/?q=ai:panizzut.marta"Sturmfels, Bernd"https://www.zbmath.org/authors/?q=ai:sturmfels.berndSummary: \(K3\) polytopes appear in complements of tropical quartic surfaces. They are dual to regular unimodular central triangulations of reflexive polytopes in the fourth dilation of the standard tetrahedron. Exploring these combinatorial objects, we classify \(K3\) polytopes with up to 30 vertices. Their number is \(36,297,333\). We study the singular loci of quartic surfaces that tropicalize to \(K3\) polytopes. These surfaces are stable in the sense of Geometric Invariant Theory.Tropical homology.https://www.zbmath.org/1460.141462021-06-15T18:09:00+00:00"Itenberg, Ilia"https://www.zbmath.org/authors/?q=ai:itenberg.ilia-v"Katzarkov, Ludmil"https://www.zbmath.org/authors/?q=ai:katzarkov.ludmil"Mikhalkin, Grigory"https://www.zbmath.org/authors/?q=ai:mikhalkin.grigory-b"Zharkov, Ilia"https://www.zbmath.org/authors/?q=ai:zharkov.iliaThis is one of the cornerstone works in tropical geometry. The main result of the paper allows one to recover the Hodge structure of a smooth projective variety via the combinatorics of its tropicalization. More precisely, denote \[{\mathrm{Log}}_t:({\mathbb C}^*)^n\to{\mathbb R}^n,\quad t>0.\] Let \({\mathcal X}\to\{0<|z|\ll1\}\) be a flat family of smooth \(k\)-dimensional complex varieties in \({\mathbb P}^n\). It is known that there is a sequence \(t_i\underset{i\to\infty}{\longrightarrow}0\), \(t_i>0\), such that the sequence \({\mathrm{Log}}_{t_i}({\mathcal X}_{t_i}\cap({\mathbb C}^*)^n)\) converges in Hausdorff topology to a \(k\)-dimensional tropical vriety \(V\subset{\mathbb R}^n\), which is a finite, rational, weighted, balanced polyhedral complex of pure dimension \(k\) (called the tropicalization of the family \({\mathcal X}_t\)). For a cell \(\tau\) of \(V\), let \({}^{\mathbb Z}{\mathcal F}_p(\tau)\subset\Lambda^p{\mathbb Z}^n\) be the subgroup generated by all elements \(v_1\wedge\dots\wedge v_p\), where \(v_1,\dots,v_p\in T_\sigma\cap{\mathbb Z}^n\), \(T_\sigma\) being the tangent space to a cell \(\sigma\supset\tau\). Since \({}^{\mathbb Z}{\mathcal F}_p(\tau)\subset{}^{\mathbb Z}{\mathcal F}_p(\tau')\) as long as \(\tau\subset\tau'\), one can define the tropical homology groups \[H_{p,q}(V)=H_q(V,{}^{\mathbb Z}{\mathcal F}_p\otimes{\mathbb Q}),\quad p,q\ge0.\] The main result of the paper states that, if \(V\) is smooth (in the tropical sense), that the groups \(H_{p,q}(V)\) are naturally isomorphic to \(W_{2p}/W_{2p-1}\), where \(W_m\) denotes the \(m\)-th graded piece of the monodromy weight filtration in the mixed Hodge structure of \(H^{p+q}({\mathcal Z}_\infty,{\mathbb Q})\) with a canonical fiber \({\mathcal X}_\infty\) of the family \({\mathcal X}\). In particular, for a generic fiber \({\mathcal Z}_t\), one has \(h^{p,q}({\mathcal Z}_t)=\dim_{\mathbb Q}H_{p,q}(V)\). The key ingrediant of the proof is a quasi-isomorphism between the tropical cellular complexes and the dual row complexes of the the page \(E^1\) of the weight spectral sequence for the limiting mixed Hodge structure of \({\mathcal Z}_\infty\).
Reviewer: Eugenii I. Shustin (Tel Aviv)Topology of tropical moduli of weighted stable curves.https://www.zbmath.org/1460.141452021-06-15T18:09:00+00:00"Cerbu, Alois"https://www.zbmath.org/authors/?q=ai:cerbu.alois"Marcus, Steffen"https://www.zbmath.org/authors/?q=ai:marcus.steffen"Peilen, Luke"https://www.zbmath.org/authors/?q=ai:peilen.luke"Ranganathan, Dhruv"https://www.zbmath.org/authors/?q=ai:ranganathan.dhruv"Salmon, Andrew"https://www.zbmath.org/authors/?q=ai:salmon.andrewSummary: The moduli space \(\Delta_{g,w}\) of tropical \(w\)-weighted stable curves of volume 1 is naturally identified with the dual complex of the divisor of singular curves in Hassett's spaces of \(w\)-weighted stable curves. If at least two of the weights are 1, we prove that \(\Delta_{0, w}\) is homotopic to a wedge sum of spheres, possibly of varying dimensions. Under additional natural hypotheses on the weight vector, we establish explicit formulas for the Betti numbers of the spaces. We exhibit infinite families of weights for which the space \(\Delta_{0,w}\) is disconnected and for which the fundamental group of \(\Delta_{0,w}\) has torsion. In the latter case, the universal cover is shown to have a natural modular interpretation. This places the weighted variant of the space in stark contrast to the heavy/light cases studied previously by \textit{K. Vogtmann} [Proc. Edinb. Math. Soc., II. Ser. 33, No. 3, 367--379 (1990; Zbl 0694.20021)] and \textit{R. Cavalieri} et al. [Forum Math. Sigma 4, Paper No. e9, 35 p. (2016; Zbl 1373.14063)]. Finally, we prove a structural result relating the spaces of weighted stable curves in genus 0 and 1, and leverage this to extend several of our genus 0 results to the spaces \(\Delta_{1,w}\).