## 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?

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.