Kris Shaw

• Aksnes, Edvard; Amini, Omid; Piquerez, Matthieu & Shaw, Kris (2025). COHOMOLOGICALLY TROPICAL VARIETIES. Journal of the Institute of Mathematics of Jussieu. ISSN 1474-7480. 24(6), p. 2572–2572. doi: 10.1017/s1474748025101114. Full text in Research Archive Show summary

  Given the tropicalization of a complex subvariety of the torus, we define a morphism between the tropical cohomology and the rational cohomology of their respective tropical compactifications. We say that the subvariety of the torus is cohomologically tropical if this map is an isomorphism for all closed strata of the tropical compactification. We prove that a schön subvariety of the torus is cohomologically tropical if and only if it is wunderschön and its tropicalization is a tropical homology manifold. The former property means that the open strata in the boundary of a tropical compactification are all connected and the mixed Hodge structures on their cohomology are pure of maximum possible weight; the latter property requires that, locally, the tropicalization verifies tropical Poincaré duality. We study other properties of cohomologically tropical and wunderschön varieties, and show that in a semistable degeneration to an arrangement of cohomologically tropical varieties, the Hodge numbers of the smooth fibers are captured in the tropical cohomology of the tropicalization. This extends the results of Itenberg, Katzarkov, Mikhalkin and Zharkov.

• Rau, Johannes; Renaudineau, Arthur & Shaw, Kris (2025). Real Phase Structures on Tropical Varieties and Patchworks in Higher Codimension. International Mathematics Research Notices (IMRN). ISSN 1073-7928. 2025(20). doi: 10.1093/imrn/rnaf312. Full text in Research Archive
• Markwig, Hannah; Payne, Sam & Shaw, Kris (2023). Bitangents to plane quartics via tropical geometry: rationality, A^1-enumeration, and real signed count. Research in the Mathematical Sciences. ISSN 2522-0144. 10(2). doi: 10.1007/s40687-023-00383-1. Full text in Research Archive
• Shaw, Kris & Werner, Annette (2023). ON THE BIRATIONAL GEOMETRY OF MATROIDS. Combinatorial Theory. 3(2). doi: 10.5070/C63261996. Full text in Research Archive
• Kastner, Lars; Shaw, Kris & Winz, Anna-Lena (2023). Tropical Compactification via Ganter’s Algorithm. Discrete & Computational Geometry. ISSN 0179-5376. doi: 10.1007/s00454-023-00581-2. Full text in Research Archive Show summary

  Abstract We describe a canonical compactification of a polyhedral complex in Euclidean space. When the recession cones of the polyhedral complex form a fan, the compactified polyhedral complex is a subspace of a tropical toric variety. In this case, the procedure is analogous to the tropical compactifications of subvarieties of tori. We give an analysis of the combinatorial structure of the compactification and show that its Hasse diagram can be computed via Ganter’s algorithm. Our algorithm is implemented in and shipped with .

• Markwig, Hannah; Payne, Sam & Shaw, Kris (2023). Avoidance loci and tropicalizations of real bitangents to plane quartics. Proceedings of the Royal Society of Edinburgh. Section A Mathematics. ISSN 0308-2105. doi: 10.1017/prm.2023.67. Full text in Research Archive Show summary

  We compare two partitions of real bitangents to smooth plane quartics into sets of 4: one coming from the closures of connected components of the avoidance locus and another coming from tropical geometry. When both are defined, we use the Tarski principle for real closed fields in combination with the topology of real plane quartics and the tropical geometry of bitangents and theta characteristics to show that they coincide.

• Renaudineau, Arthur & Shaw, Kris (2023). Bounding the betti numbers of real hypersurfaces near the tropical limit. Annales Scientifiques de l'Ecole Normale Supérieure. ISSN 0012-9593. 56(3), p. 945–980. doi: 10.24033/asens.2547. Full text in Research Archive Show summary

  A bound is demonstrated by Itenberg on the Betti numbers of real algebraic hypersurfaces close to the tropical boundary. These bounds are expressed as a function of the Hodge numbers of the complexification. To demonstrate these bounds, we introduce a variant of tropical homology into the real framework, and define filtration on the associated chain complex, inspired by Kalinin filtration. The spectral sequence associated with this filtration converges towards the homology groups of the real algebraic variety, and we show that the terms of the first page are the groups of tropical homology (with coefficients in2 / 2Z). The dimensions of these homology groups correspond to the Hodge numbers of the complex projective hypersurfaces thanks to the results of Itenberg, Mikhalkin, Katzarkov, on the one hand, and authors with Arnal on the other. The bounds on the Betti numbers enmentate, as well as a criterion for obtaining a maximum variety. We also generalize the formula due to Bertrand linking the signature of a complex hypersurface and the Euler characteristic of a real hypersurface, and we will redequip the combinatorial criterion of Haas on the maximum of the flat curves close to the tropical boundary.

• Rau, Johannes; Renaudineau, Arthur & Shaw, Kris (2022). Real phase structures on matroid fans and matroid orientations. Journal of the London Mathematical Society. ISSN 0024-6107. 106(4), p. 3687–3710. doi: 10.1112/jlms.12671. Full text in Research Archive Show summary

  We introduce the notion of real phase structure on rational polyhedral fans in Euclidean space. Such a structure consists of an assignment of affine spaces over to each top-dimensional face of the fan subject to two conditions. Given an oriented matroid we can construct a real phase structure on the fan of the underlying matroid. Conversely, we show that from a real phase structure on a matroid fan we can produce an orientation of the underlying matroid. Thus real phase structures are cryptomorphic to matroid orientations. The topes of the orientated matroid are recovered immediately from the real phase structure. We also provide a direct way to recover the signed circuits of the oriented matroid from the real phase structure.

• Markwig, Hannah; Markwig, Thomas; Shaw, Kris & Shustin, Eugenii (2022). Tropical floor plans and enumeration of complex and real multi-nodal surfaces. Journal of Algebraic Geometry. ISSN 1056-3911. 31(2), p. 261–301. doi: 10.1090/jag/774. Full text in Research Archive
• Arnal, Charles; Renaudineau, Arthur & Shaw, Kris (2021). Lefschetz section theorems for tropical hypersurfaces. Annales Henri Lebesgue (AHL). 4, p. 1347–1387. doi: 10.5802/ahl.104. Full text in Research Archive Show summary

  We establish variants of the Lefschetz section theorem for the integral tropical homology groups of tropical hypersurfaces of tropical toric varieties. It follows from these theorems that the integral tropical homology groups of non-singular tropical hypersurfaces which are compact or contained in ℝ n are torsion free. We prove a relationship between the coefficients of the χ y genera of complex hypersurfaces in toric varieties and Euler characteristics of the integral tropical cellular chain complexes of their tropical counterparts. It follows that the integral tropical homology groups give the Hodge numbers of compact non-singular hypersurfaces of complex toric varieties. Finally for tropical hypersurfaces in certain affine toric varieties, we relate the ranks of their tropical homology groups to the Hodge–Deligne numbers of their complex counterparts.

• Medrano, Lucia López de; Rincon, Felipe & Shaw, Kristin (2020). Chern–Schwartz–MacPherson cycles of matroids. Proceedings of the London Mathematical Society. ISSN 0024-6115. 120(1), p. 1–27. doi: 10.1112/plms.12278. Full text in Research Archive
• Kummer, Mario & Shaw, Kristin (2020). The separating semigroup of a real curve. Annales de la Faculté des Sciences de Toulouse (AFST). ISSN 0240-2963. 29(1), p. 79–96. doi: 10.5802/afst.1624. Full text in Research Archive Show summary

  We introduce the separating semigroup of a real algebraic curve of dividing type. The elements of this semigroup record the possible degrees of the covering maps obtained by restricting separating morphisms to the real part of the curve. We also introduce the hyperbolic semigroup which consists of elements of the separating semigroup arising from morphisms which are compositions of a linear projection with an embedding of the curve to some projective space. We completely determine both semigroups in the case of maximal curves. We also prove that any embedding of a real curve of dividing type to projective space of sufficiently high degree is hyperbolic. Using these semigroups we show that the hyperbolicity locus of an embedded curve is in general not connected.

• Mohammadi, Fatemeh & Shaw, Kristin (2019). Toric degenerations of Grassmannians from matching fields. Algebraic combinatorics. 2(6), p. 1109–1124. doi: 10.5802/alco.77. Full text in Research Archive Show summary

  We study the algebraic combinatorics of monomial degenerations of Plücker forms which is governed by matching fields in the sense of Sturmfels and Zelevinsky. We provide a necessary condition for a matching field to yield a SAGBI basis of the Plücker algebra for 3-planes in n-space. When the ideal associated to the matching field is quadratically generated this condition is both necessary and sufficient. Finally, we describe a family of matching fields, called 2-block diagonal, whose ideals are quadratically generated. These matching fields produce a new family of toric degenerations of Gr(3,n).

• Shaw, Kristin; Brugallé, Erwan; Mikhalkin, Grigory & Risler, Jean-Jacques (2018). Nonexistence of torically maximal hypersurfaces. Алгебра и анализ. ISSN 0234-0852. Full text in Research Archive
• Jell, Philipp; Rau, Johannes & Shaw, Kristin (2018). Lefschetz (1,1) theorem in tropical geometry. Épijournal de Géométrie Algébrique (EPIGA). Full text in Research Archive Show summary

  For a tropical manifold of dimension n we show that the tropical homology classes of degree (n-1, n-1) which arise as fundamental classes of tropical cycles are precisely those in the kernel of the eigenwave map. To prove this we establish a tropical version of the Lefschetz (1, 1)-theorem for rational polyhedral spaces that relates tropical line bundles to the kernel of the wave homomorphism on cohomology. Our result for tropical manifolds then follows by combining this with Poincaré duality for integral tropical homology.

• Kastner, Lars; Shaw, Kristin & Winz, Anna-Lena (2017). Cellular sheaf cohomology of polymake. Fields Institute Communications. ISSN 1069-5265. p. 369–385. doi: 10.1007/978-1-4939-7486-3_17. Full text in Research Archive
• Shaw, Kristin; Ren, Qinchung & Sturmfels, Bernd (2016). Tropicalizations of del Pezzo surfaces. Advances in Mathematics. ISSN 0001-8708. Full text in Research Archive
• Shaw, Kristin & Brugallé, Erwan (2015). Obstructions to approximating tropical curves in surfaces via intersection theory. Canadian Journal of Mathematics (CJM). ISSN 0008-414X. doi: 10.4153/cjm-2014-014-4. Full text in Research Archive Show summary

  We provide some new local obstructions to approximating tropical curves in smooth tropical surfaces. These obstructions are based on a relation between tropical and complex intersection theories, which is also established here. We give two applications of the methods developed in this paper. First we classify all locally irreducible approximable 3-valent fan tropical curves in a fan tropical plane. Secondly, we prove that a generic non-singular tropical surface in tropical projective 3-space contains finitely many approximable tropical lines if it is of degree 3, and contains no approximable tropical lines if it is of degree 4 or more.

• Shaw, Kristin & Brugallé, Erwan (2014). A bit of tropical geometry. The American Mathematical Monthly. ISSN 0002-9890. doi: 10.4169/amer.math.monthly.121.07.563. Full text in Research Archive Show summary

  This friendly introduction to tropical geometry is meant to be accessible to first year students in mathematics. The topics discussed here are basic tropical algebra, tropical plane curves, some tropical intersections, and Viro's patchworking. Each definition is explained with concrete examples and illustrations. The text is a modification of a translation from a French text by the first author. There is also a newly-added section highlighting new developments and perspectives on tropical geometry. In addition, the final section provides an extensive list of references on the subject.

• Shaw, Kristin (2013). A tropical intersection product in matroidal fans. SIAM Journal on Discrete Mathematics. ISSN 0895-4801. doi: 10.1137/110850141. Full text in Research Archive Show summary

  We construct an intersection product on tropical cycles contained in the Bergman fan of a matroid. To do this we first establish a connection between the operations of deletion and restriction in matroid theory and tropical modifications as defined by Mikhalkin in [Proceedings of the International Congress of Mathematicians, Vol. II, European Mathematical Society, Zürich, 2006, pp. 827--852]. This product generalizes the product of Allermann and Rau [Math. Z., 264 (2010), pp. 633--670] and Allermann [Tropical intersection products on smooth varieties, J. Eur. Math. Soc. (JEMS), 14 (2012), pp. 107--126] and also provides an alternative procedure for intersecting cycles which is not based on intersecting with Cartier divisors. Also, we simplify the definition in the case of one-dimensional fan cycles in two-dimensional matroidal fans and give an application of the intersection product to realiability questions in tropical geometry.

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