We discuss chemical reaction networks and metabolic pathways based on stoichiometric network analysis, and introduce deformed toric ideal constraints by the algebraic geometrical approach. With the deformed toric ideal constraints, the shape of flux is constrained without introducing ad hoc constraints. To illustrate the effectiveness of such constraints, we discuss two examples of chemical rea...

These are introductory lecture notes on complex geometry, Calabi–Yau manifolds and toric geometry. We first define basic concepts of complex and Kähler geometry. We then proceed with an analysis of various definitions of Calabi–Yau manifolds. The last section provides a short introduction to toric geometry, aimed at constructing Calabi–Yau manifolds in two different ways; as hypersurfaces in to...

We present a self-contained combinatorial approach to Fujita’s conjectures in the toric case. Our main new result is a generalization of Fujita’s very ampleness conjecture for toric varieties with arbitrary singularities. In an appendix, we use similar methods to give a new proof of an analogous toric generalization of Fujita’s freeness conjecture due to Fujino.

We discuss a characteristic free version of Frobenius splittings for toric varieties and give a polyhedral criterion for a toric variety to be diagonally split. We apply this criterion to show that section rings of nef line bundles on diagonally split toric varieties are normally presented and Koszul, and that Schubert varieties are not diagonally split in general.

We plan to discuss how the ideas and methodology of Toric Topology can be applied to one of the classical subjects of algebraic topology: finding nice representatives in complex cobordism classes. Toric and quasitoric manifolds are the key players in the emerging field of Toric Topology, and they constitute a sufficiently wide class of stably complex manifolds to additively generate the whole c...

Algebraic Geometry is a deep subject of remarkable power that originated in the study of concrete objects, as it is concerned with the geometry of solutions to polynomial equations. It is this interplay between theoretical abstraction and tangible examples that gives the subject relevance and a reach into other areas of mathematics and its applications. There is no better subdiscipline of algeb...

The paradigmatic result in symplectic toric geometry is the paper of Delzant that classifies compact connected symplectic manifolds with effective completely integrable torus actions, the so called (compact) symplectic toric manifolds. The moment map induces an embedding of the quotient of the manifold by the torus action into the dual of the Lie algebra of the torus; its image is a simple unim...

Long linear codes constructed from toric varieties over finite fields, their multiplicative structure and decoding. The main theme is the inherent multiplicative structure on toric codes. The multiplicative structure allows for decoding, resembling the decoding of Reed-Solomon codes and aligns with decoding by error correcting pairs. We have used the multiplicative structure on toric codes to c...

Let (P∆, ω) be a toric variety whose moment map image (with respect to the toric Kähler form ω) is the real convex polyhedron ∆ ⊂ MR. Also assume that the anti-canonical class of P∆ is represented by an integral reflexive convex polyhedron ∆0 ⊂ M and the unique interior point of ∆0 is the origin of M . Integral points m ∈ ∆0 correspond to holomorphic toric sections sm of the anticanonical bundl...

We combine work of Cox on the homogeneous coordinate ring of a toric variety and results of Eisenbud-Mustaţǎ-Stillman and Mustaţǎ on cohomology of toric and monomial ideals to obtain a formula for computing χ(OX (D)) for a divisor D on a complete simplicial toric variety XΣ. The main point is to use Alexander duality to pass from the toric irrelevant ideal, which appears in the computation of χ...