Related papers: Linear precision for toric surface patches
We give a precise mathematical formulation for the notions of a parametric patch and linear precision, and establish their elementary properties. We relate linear precision to the geometry of a particular linear projection, giving necessary…
We give a general criterion for two toric varieties to appear as fibers in a flat family over the projective line. We apply this to show that certain birational transformations mapping a Laurent polynomial to another Laurent polynomial…
The affine space of all tensor product B\'ezier patches of degree nxn with prescribed main diagonal curves is determined. First, the pair of B\'ezier curves which can be diagonals of a B\'ezier patch is characterized. Besides prescribing…
The control polygon of a Bezier curve is well-defined and has geometric significance---there is a sequence of weights under which the limiting position of the curve is the control polygon. For a Bezier surface patch, there are many possible…
We classify real Poisson structures on complex toric manifolds of type $(1,1)$ and initiate an investigation of their Poisson cohomology. For smooth toric varieties, such structures are necessarily algebraic and are homogeneous quadratic in…
We consider linear systems on toric varieties of any dimension, with invariant base points, giving a characterization of special linear systems. We then make a new conjecture for linear systems on rational surfaces.
Given a hypersurface in a complex projective space, we prove that the multidegrees of its toric polar map agree, up to sign, with the coefficients of the Chern-Schwartz-MacPherson class of a distinguished open set, namely the complement of…
Rational B\'{e}zier functions are widely used as mapping functions in surface reparameterization, finite element analysis, image warping and morphing. The injectivity (one-to-one property) of a mapping function is typically necessary for…
We show that a weight variety, which is a quotient of a flag variety by the maximal torus, admits a flat degeneration to a toric variety. In particular, we show that the moduli spaces of spatial polygons degenerate to polarized toric…
The Losev-Manin moduli space parametrizes pointed chains of projective lines. In this paper we study a possible generalization to families of pointed degenerate toric varieties. Geometric properties of these families, such as flatness and…
Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this thesis we explore this correspondence to classify smooth lattice…
In this paper, we give a description of holomorphic multi-vector fields on smooth compact toric varieties, which generalizes Demazure's result of holomorphic vector fields on toric varieties. Based on the result, we compute the Poisson…
In this note we study linear systems on complete toric varieties $X$ with an invariant point, whose orbit under the action of the automorphism group of $X$ contains the dense torus $T$ of $X$. We give a characterization of such varieties in…
In this paper we show that a surface in P^3 parametrized over a 2-dimensional toric variety T can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection. This…
We consider the derived category of permutation modules for a finite group, in positive characteristic. We stratify this tensor triangulated category using Brauer quotients. We describe the spectrum of its compact objects, by reducing the…
We formulate a conjecture which describes the Fukaya category of an exact Lefschetz fibration defined by a Laurent polynomial in two variables in terms of a pair consisting of a consistent dimer model and a perfect matching on it. We prove…
In [1], the author considered the problem of the optimal approximation of symmetric surfaces by biquadratic B\'ezier patches. Unfortunately, the results therein are incorrect, which is shown in this paper by considering the optimal…
We call a sheaf on an algebraic variety immaculate if it lacks any cohomology including the zero-th one, that is, if the derived version of the global section functor vanishes. Such sheaves are the basic tools when building exceptional…
In this note, we make a step towards the classification of toric surfaces admitting reducible Severi varieties. We generalize the results of [Lan19, Tyo13, Tyo14], and provide two families of toric surfaces admitting reducible Severi…
Toric quiver varieties (moduli spaces of quiver representations) are studied. Given a quiver and a weight there is an associated quasiprojective toric variety together with a canonical embedding into projective space. It is shown that for a…