I am studying ideals such as toric ideals but I am unable to find a consistent definition, it seems to be very general so please explain the origin of “toric ideal”. Is there a geometric interpretation for toric ideal for example in three dimensions? Has toric ideal something to do with torus?

Let $\Delta$ be a fan, consisting of cones $\sigma_0=conv(e_1,e_1+e_2)$ and $\sigma_2=conv(e_1+e_2,e_2)$ and $\tau=\sigma_0\cap\sigma_1=conv(e_1+e_2)$. The dual cones are $\sigma_0^\vee= conv(e_2,e_1\!-\!e_2)$ and $\sigma_1^\vee= conv(e_2\!-\!e_1,e_1)$ and $\tau^\vee= conv(e_2\!-\!e_1,e_1\!-\!e_2,e_1,e_2)= conv(e_2\!-\!e_1,e_1\!-\!e_2,e_1)= conv(e_2\!-\!e_1,e_1\!-\!e_2,e_2)$. The corresponding semigroup algebras are $S_{\sigma_0}= \mathbb{C}[\mathbf{x}^{e_2},\mathbf{x}^{e_1\!-\!e_2}] =\mathbb{C}[y,xy^{-1}]$ and $S_{\sigma_1}= \mathbb{C}[\mathbf{x}^{e_2\!-\!e_1},\mathbf{x}^{e_1}]= \mathbb{C}[x^{-1}y,x]$ and $S_\tau= \mathbb{C}[\mathbf{x}^{e_2\!-\!e_1}, \mathbf{x}^{e_1\!-\!e_2}, \mathbf{x}^{e_1},\mathbf{x}^{e_2}]$ $=$ $\mathbb{C}[x^{-1}y,y^{-1}x,x,y]$ $=$ $\mathbb{C}[x^{-1}y,y^{-1}x,x]= \mathbb{C}[x^{-1}y,y^{-1}x,y]$ inside $\mathbb{C}[x^{\pm1},y^{\pm1}]$. The three associated affine […]

What is a good book on algebraic geometry, with focus on toric varieties, similar both in the philosophy and in the prestige of the autors to Modern Geometric Structures and Fields by Novikov and Taimanov ? My background is from theoretical and mathematical physics, and I need toric geometry for string theory and mirror symmetry […]

A rational convex polyehedral cone $\sigma\subseteq\mathbb R^n$ is a set of the form $$ \sigma=\operatorname{Cone}(u_1,\dots,u_k) := \left\{ \sum_{i=1}^k r_i u_i \,\Bigg|\, r_i\ge 0\right\}\subseteq\mathbb R^n,$$ where all $u_i\in\mathbb Z^n$. I’m interested in what is really involved in proving the following statements: $\sigma^\vee = \{ v\in \mathbb R^n \,|\, \langle u,v\rangle \ge 0 \,\forall u\in\sigma\}$ is a […]

Let $A$ be a graded ring. Note that the grading of $A$ may not be $\mathbb{N}$, for example, the grading of $A$ could be $\mathbb{Z}^n$. Actually, my question comes from the paper of Tamafumi’s On Equivariant Vector Bundles On An Almost Homogeneous Variety, Proposition 3.4. And I translate this proposition to modern language: Let $A […]

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