ag.algebraic geometry – Distinguishing ample divisors by minimally intersecting curves on a projective simplicial toric variety

My question has an easily formulated generalization, which I will state first. Let $sigma subseteq mathbf{R}^n$ be a strongly convex polyhedral cone. For each minimally generating lattice point $m in sigma^o cap mathbf{Z}^n$ of the interior cone $sigma^o subseteq sigma$, let $S(m) subseteq sigma^{vee} cap mathbf{Z}^n$ denote the set of lattice points $u$ with $langle u,m rangle = 1$. My question is:

Does $S(m) = S(m’)$ imply that $m = m’$?

As a special case, assume that $sigma$ is the nef cone of a simplicial projective toric variety $X_{Sigma}$. Then my question seems to amount to the following:

If $D_1$ and $D_2$ are two ample divisors minimally generating in the ample cone, then does $D_1 cdot C = 1 Leftrightarrow D_2 cdot C = 1$ for all effective curves $C$ imply that $D_1 = D_2$?

This is the case I am most interested in.

reference request – Slodowy slice intersecting a given orbit “minimally”?

Let $mathfrak{g}$ be a complex semisimple Lie algebra. Is it true that for any $Xinmathfrak{g}$, there exists an $mathfrak{sl}_2$-triple $(e,h,f)$ in $mathfrak{g}$ such that

  1. The conjugacy orbit of $X$ meets the Slodowy slice $e+Z_{mathfrak{g}}(f)$, and
  2. $dim Z_{mathfrak{g}}(X)=dim Z_{mathfrak{g}}(e)$?

(One can deduce from the above conditions that the orbit of $X$ must then meet the slice transversally, like in classical Kostant section situation.)

When $X$ is regular this is the well-known result about Kostant section. When $mathfrak{g}=mathfrak{gl}_n$ this is also true and can be deduced from the rational form of $X$. It might be that this question can be answered by a simple reference, but thanks a lot in advance in all cases!

magento2 – Magento 2 How to load the product by sku and start the system minimally?

Hi!

I need a minimum load system that gets ONLY: Product name, product type, product URL, thumbnail URL, price, configurable by-products.

Preferably use catalog_product_flat_ * table (I think it's a good idea for optimization)

How can I do it?

Please help me! Thank you!

co.combinatorics – When does a biconnected graph have a minimally strong orientation?

Given any asymmetric relationship $ A subseteq V ^ 2 $ a digraph $ D = (V, A) $ is minimally strong iff $ D $ is strongly connected and for each arc $ a in A $ the digraph $ D – a = (V, A setminus {a }) $ not strongly connected

Now for Robbins' theorem every $ 2 $The connected edge chart has a strong orientation, so clearly every biconnected chart must have a strong orientation. However, not all graphics have a minimally strong orientation, for example, wheel graphics https://en.wikipedia.org/wiki/Wheel_graph can never have a minimally strong orientation. So which biconnected graphics have a minimally strong orientation?

I can prove that if there is a biconnected graph $ G $ has a minimally strong orientation then $ chi (G) leq 3 $ as well as that $ | V (G) | leq | E (G) | leq 2 | V (G) | -3 $ also if we contract any triangle $ T subseteq G $ to form a graph $ G / T $ then this chart should also have a minimally strong orientation, similarly consider if $ G $ it has a minimally strong orientation, then each subdivision of $ G $. It is also quite simple to prove that each graph that does not have cycle chords must have a minimally strong orientation. Also the chart $ G $ It will always have a vertex with a degree equal to two.

However, I am not sure how to proceed from here, at the bottom of the second page of this document, the author briefly mentions these graphics and the article itself takes time to derive a technique to build minimally strong digraphs, although it is a Little hard to follow. , it seems that you are using a combination of clusters of two vertices and subdivisions in directed graphs defined recursively. Therefore, I believe that by studying the SPQR tree of these graphs, one might be able to derive some simple criteria for which the connected graphs have a minimally strong orientation. Although I am not familiar with all this and I wonder if this is excessive. Can anyone help me here?