arXiv subjects

Diane Maclagan

Publications and source records attributed to Diane Maclagan.

At least 19 recordsLinked to original sources

arXiv2024

Tropical Vector Bundles

The goal of this paper is to introduce a construction of a vector bundle on a tropical variety. When the base is a tropical toric variety these tropicalize toric vector bundles, and are described by the data of a valuated matroid and some flats in the lattice of flats of the underlying matroid. The fibers are tropical linear spaces. We define global sections for tropical toric vector bundles, stability, and Jordan-Hölder and Harder-Narasimhan filtrations. Many of these require additional modularity assumptions on the defining matroid.

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arXiv2023

The spine of the T-graph of the Hilbert scheme of points in the plane

The torus T of projective space also acts on the Hilbert scheme of subschemes of projective space. The T-graph of the Hilbert scheme has vertices the fixed points of this action, and edges connecting pairs of fixed points in the closure of a one-dimensional orbit. In general this graph depends on the underlying field. We construct a subgraph, which we call the spine, of the T-graph of Hilb^m(A^2) that is independent of the choice of infinite field. For certain edges in the spine we also give a description of the tropical ideal, in the sense of tropical scheme theory, of a general ideal in the edge. This gives a more refined understanding of these edges, and of the tropical stratification of the Hilbert scheme.

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arXiv2021

Toric and tropical Bertini theorems in positive characteristic

We generalize the toric Bertini theorem of Fuchs, Mantova, and Zannier to positive characteristic. A key part of the proof is a new algebraically closed field containing the field \kk(t_1,\dots,t_d) of rational functions over an algebraically closed field \kk of prime characteristic. As a corollary, we extend the tropical Bertini theorem of Maclagan and Yu to arbitrary characteristic, which removes the characteristic dependence from the d-connectivity result for tropical varieties from that paper.

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arXiv2021

Higher Connectivity of Tropicalizations

We show that the tropicalization of an irreducible d-dimensional variety over a field of characteristic 0 is (d-l)-connected through codimension one, where l is the dimension of the lineality space of the tropicalization. From this we obtain a higher connectivity result for skeleta of rational polytopes. We also prove a tropical analogue of the Bertini Theorem: the intersection of the tropicalization of an irreducible variety with a generic hyperplane is again the tropicalization of an irreducible variety.

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arXiv2020

Varieties of Tropical Ideals are Balanced

Tropical ideals, introduced in arXiv:1609.03838, define subschemes of tropical toric varieties. We prove that the top-dimensional parts of their varieties are balanced polyhedral complexes of the same dimension as the ideal. This means that every subscheme of a tropical toric variety defined by a tropical ideal has an associated class in the Chow ring of the toric variety. A key tool in the proof is that specialization of variables in a tropical ideal yields another tropical ideal; this plays the role of hyperplane sections in the theory. We also show that elimination theory (projection of varieties) works for tropical ideals as in the classical case. The matroid condition that defines tropical ideals is crucial for these results.

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arXiv2019

Computing Tropical Varieties in Macaulay2

We introduce a package for doing tropical computations in Macaulay2. The package draws on the functionality of Gfan and Polymake while making the process as simple as possible for the end user. This provides a powerful and user friendly tool for computing tropical varieties requiring little prerequisite knowledge.

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arXiv2017

A presentation for the Cox ring of $\overline{M}_{0,6}$

We compute the relations in the Cox ring of the moduli space $\overline{M}_{0,6}$. This gives a presentation of the Cox ring as a quotient of a polynomial ring with 40 generators by an ideal with 225 generators that come in 5 symmetry classes.

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arXiv2017

Tropical schemes, tropical cycles, and valuated matroids

We show that the weights on a tropical variety can be recovered from the tropical scheme structure proposed by the Giansiracusas in arXiv:1308.0042, so there is a well-defined Hilbert-Chow morphism from a tropical scheme to the underlying tropical cycle. For a subscheme of projective space given by a homogeneous ideal I we show that this tropical scheme structure contains the same information as the set of valuated matroids of the vector spaces I_d for d \geq 0. We also give a combinatorial criterion to determine whether a given relation is in the congruence defining the tropical scheme structure.

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arXiv2017

Tropical Ideals

We introduce and study a special class of ideals, called tropical ideals, in the semiring of tropical polynomials, with the goal of developing a useful and solid algebraic foundation for tropical geometry. The class of tropical ideals strictly includes the tropicalizations of classical ideals, and allows us to define subschemes of tropical toric varieties, generalizing [Giansiracusa-Giansiracusa]. We investigate some of the basic structure of tropical ideals, and show that they satisfy many desirable properties that mimic the classical setup. In particular, every tropical ideal has an associated variety, which we prove is always a finite polyhedral complex. In addition we show that tropical ideals satisfy the ascending chain condition, even though they are typically not finitely generated, and also the weak Nullstellensatz.

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arXiv2017

Groebner bases over fields with valuations

Let K be a field with a valuation and let S be the polynomial ring S:= K[x_1,..., x_n]. We discuss the extension of Groebner theory to ideals in S, taking the valuations of coefficients into account, and describe the Buchberger algorithm in this context. In addition we discuss some implementation and complexity issues. The main motivation comes from tropical geometry, as tropical varieties can be defined using these Groebner bases, but we also give examples showing that the resulting Groebner bases can be substantially smaller than traditional Groebner bases. In the case K = Q with the p-adic valuation the algorithms have been implemented in a Macaulay 2 package.

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arXiv2013

Polyhedral structures on tropical varieties

Tropical varieties are polyhedral shadows of classical varieties. The purpose of these expository notes is to explain the origin of this polyhedral complex structure from the perspective of Gröbner bases. To appear in the proceedings of the 2011 Bellairs Workshop in Number Theory.

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arXiv2012

Introduction to tropical algebraic geometry

This is an expository introduction to tropical algebraic geometry based on my lectures at the Workshop on Tropical Geometry and Integrable Systems in Glasgow, July 4-8, 2011, and at the ELGA 2011 school on Algebraic Geometry and Applications in Buenos Aires, August 1-5, 2011.

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arXiv2011

The T-graph of a multigraded Hilbert scheme

The T-graph of a multigraded Hilbert scheme records the zero and one-dimensional orbits of the T = (K^*)^n action on the Hilbert scheme induced from the T-action on A^n. It has vertices the T-fixed points, and edges the one-dimensional T-orbits. We give a combinatorial necessary condition for the existence of an edge between two vertices in this graph. For the Hilbert scheme of points in the plane, we give an explicit combinatorial description of the equations defining the scheme parameterizing all one-dimensional torus orbits whose closures contain two given monomial ideals. For this Hilbert scheme we show that the T-graph depends on the ground field, resolving a question of Altmann and Sturmfels.

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arXiv2011

Lower and upper bounds for nef cones

The nef cone of a projective variety Y is an important and often elusive invariant. In this paper we construct two polyhedral lower bounds and one polyhedral upper bound for the nef cone of Y using an embedding of Y into a toric variety. The lower bounds generalize the combinatorial description of the nef cone of a Mori dream space, while the upper bound generalizes the F-conjecture for the nef cone of the moduli space \bar{M}_{0,n} to a wide class of varieties.

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arXiv2009

Smooth and irreducible multigraded Hilbert schemes

The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial ring is ZZ[x,y], which establishes a conjecture of Haiman and Sturmfels.

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arXiv2009

Equations for Chow and Hilbert Quotients

We give explicit equations for the Chow and Hilbert quotients of a projective scheme X by the action of an algebraic torus T in an auxiliary toric variety. As a consequence we provide GIT descriptions of these canonical quotients, and obtain other GIT quotients of X by variation of GIT quotient. We apply these results to find equations for the moduli space \bar{M}_{0,n} of stable genus zero n-pointed curves as a subvariety of a smooth toric variety defined via tropical methods.

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arXiv2007

Moduli of McKay quiver representations II: Groebner basis techniques

In this paper we introduce several computational techniques for the study of moduli spaces of McKay quiver representations, making use of Groebner bases and toric geometry. For a finite abelian group G in GL(n,k), let Y_θbe the coherent component of the moduli space of θ-stable representations of the McKay quiver. Our two main results are as follows: we provide a simple description of the quiver representations corresponding to the torus orbits of Y_θ, and, in the case where Y_θequals Nakamura's G-Hilbert scheme, we present explicit equations for a cover by local coordinate charts. The latter theorem corrects the first result from [Nakamura]. The techniques introduced here allow experimentation in this subject and give concrete algorithmic tools to tackle further open questions. To illustrate this point, we present an example of a nonnormal G-Hilbert scheme, thereby answering a question raised by Nakamura.

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arXiv2006

Moduli of McKay quiver representations I: the coherent component

For a finite abelian group G in GL(n,k), we describe the coherent component Y_theta of the moduli space M_theta of theta-stable McKay quiver representations. This is a not-necessarily-normal toric variety that admits a projective birational morphism to A^n/G obtained by variation of GIT quotient. As a special case, this gives a new construction of Nakamura's G-Hilbert scheme that avoids the (typically highly singular) Hilbert scheme of |G|-points in A^n. To conclude, we describe the toric fan of Y_theta and hence calculate the quiver representation corresponding to any point of Y_theta.

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