Toric variety

About: Toric variety is a research topic. Over the lifetime, 2630 publications have been published within this topic receiving 65604 citations. The topic is also known as: torus embedding.

Projective toric varieties as fine moduli spaces of quiver representations

Abstract: This paper proves that every projective toric variety is the fine moduli space for stable representations of an appropriate bound quiver. To accomplish this, we study the quiver $Q$ with relations $R$ corresponding to the finite-dimensional algebra $\mathop{\rm End} olimits( \textstyle\bigoplus olimits_{i=0}^{r} L_i )$ where ${\cal L} := ({\scr O}_X,L_1, \ldots, L_r)$ is a list of line bundles on a projective toric variety $X$. The quiver $Q$ defines a smooth projective toric variety, called the multilinear series $|{\cal L}|$, and a map $X \longrightarrow |{\cal L}|$. We provide necessary and sufficient conditions for the induced map to be a closed embedding. As a consequence, we obtain a new geometric quotient construction of projective toric varieties. Under slightly stronger hypotheses on ${\cal L}$, the closed embedding identifies $X$ with the fine moduli space of stable representations for the bound quiver $(Q,R)$.

Parameterizing tropical curves, I: Curves of genus zero and one

Abstract: In tropical geometry, given a curve in a toric variety, one defines a corresponding graph embedded in Euclidean space. We study the problem of reversing this process for curves of genus zero and one. Our methods focus on describing curves by parameterizations, not by their defining equations; we give parameterizations by rational functions in the genus-zero case and by nonarchimedean elliptic functions in the genus-one case. For genus-zero curves, those graphs which can be lifted can be characterized in a completely combinatorial manner. For genus-one curves, we show that certain conditions identified by Mikhalkin are sufficient and we also identify a new necessary condition.

Mirror Symmetry in Two Steps: A–I–B

Abstract: We suggest an interpretation of mirror symmetry for toric varieties via an equivalence of two conformal field theories. The first theory is the twisted sigma model of a toric variety in the infinite volume limit (the A–model). The second theory is an intermediate model, which we call the I–model. The equivalence between the A–model and the I–model is achieved by realizing the former as a deformation of a linear sigma model with a complex torus as the target and then applying to it a version of the T–duality. On the other hand, the I–model is closely related to the twisted Landau-Ginzburg model (the B–model) that is mirror dual to the A–model. Thus, the mirror symmetry is realized in two steps, via the I–model. In particular, we obtain a natural interpretation of the superpotential of the Landau-Ginzburg model as the sum of terms corresponding to the components of a divisor in the toric variety. We also relate the cohomology of the supercharges of the I–model to the chiral de Rham complex and the quantum cohomology of the underlying toric variety.

Weight systems for toric calabi-yau varieties and reflexivity of newton polyhedra

Abstract: According to a recently proposed scheme for the classification of reflexive polyhedra, weight systems of a certain type play a prominent role. These weight systems are classified for the cases n = 3 and n = 4, corresponding to toric varieties with K3 and Calabi-Yau hypersurfaces, respectively. For n = 3 we find the well-known 95 weight systems corresponding to weighted ℙ3’s that allow transverse polynomials, whereas for n = 4 there are 184,026 weight systems, including the 7555 weight systems for weighted ℙ4’s. It is proven (without computer) that the Newton polyhedra corresponding to all these weight systems are reflexive.

Bounded Negativity and Arrangements of Lines

Abstract: This thesis consists of six papers in algebraic geometry –all of which have close connections to combinatorics. In Paper A we consider complete smooth toric embeddings X ↪ P^N such that for a fixed positive integer k the t-th osculating space at every point has maximal dimension if and only if t ≤ k. Our main result is that this assumption is equivalent to that X ↪ P^N is associated to a Cayley polytope of order k having every edge of length at least k. This result generalizes an earlier characterisation by David Perkinson. In addition we prove that the above assumptions are equivalent to requiring that the Seshadri constant is exactly k at every point of X, generalizing a result of Atsushi Ito. In Paper B we introduce H-constants that measure the negativity of curves on blow-ups of surfaces. We relate these constants to the bounded negativity conjecture. Moreover we provide bounds on H-constants when restricting to curves which are a union of lines in the real or complex projective plane. In Paper C we study Gauss maps of order k for k > 1, which maps a point on a variety to its k-th osculating space at that point. Our main result is that as in the case k = 1, the higher order Gauss maps are finite on smooth varieties whose k-th osculating space is full-dimensional everywhere. Furthermore we provide convex geometric descriptions of these maps in the toric setting. In Paper D we classify fat point schemes on Hirzebruch surfaces whose initial sequence are of maximal or close to maximal length. The initial degree and initial sequence of such schemes are closely related to the famous Nagata conjecture. In Paper E we introduce the package LatticePolytopes for Macaulay2. The package extends the functionality of Macaulay2 for compuations in toric geometry and convex geometry. In Paper F we compute the Seshadri constant at a general point on smooth toric surfaces satisfying certain convex geometric assumptions on the associated polygons. Our computations relate the Seshadri constant at the general point with the jet seperation and unnormalised spectral values of the surfaces at hand.