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.

algebraic geometry – Hyperplane containing union of two intersecting lines in $mathbb P^3$

Let $l_1, l_2$ be two lines in $mathbb P^3$(over $mathbb C$) such that they intersect. Then is it always possible to obtain a hyperplane containing both of them?

Does anything like hyperplane generated by $l_1 cup l_2$ and a point outside it make any sense in this case? ( Does the fact which says that a plane contains a 2 dimensional family of lines on it contributes anything in this case?)

Can someone give me a concrete example?

Any help from anyone is welcome.

Spanning tree in a graph of intersecting sets

Consider $n$ sets, $X_i$, each having $n$ elements or fewer, drawn among a set of at most $m gt n$ elements. In other words
$$forall i in (1 ldots n),~|X_i| le n~wedge~left|bigcup_{i=1}^n X_iright| le m$$

Consider the complete graph $G$ formed by taking every $X_i$ as a node, and weighing every edge $(i,j)$ by the cardinal of the symmetric difference $X_i triangle X_j$.

An immediate bound on the weight of the minimal spanning tree is $mathcal{O}(n^2)$, since each edge is at most $2 n$, but can we refine this to $mathcal{O}(m)$?

For illustration, consider $2 p$ sets, $p$ of which contain the integers between $1$ and $p$ and $p$ of which contain the integers of between $p+1$ and $2p$. A minimal spanning tree has weight $p$ but a poorly chose tree on this graph would have weight $(p-1)p$. Intuitively, if there are only $m$ values to chose from, the sets can’t all be that different from one another.

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!

list manipulation – Generalizing previous question: Partition $[a,b]$ into $m$ sub-intervals and count number of sub-interval intersecting $A_1subseteq[a,b]$?

In my previous question, How to partition $(a,b)$ into $m$ equal sub-intervals and count the number of sub-intervals that intersect with a subset of $(a,b)$?, I was given an answer to my specific example but I’m not sure how to apply it to arbitrary $A_1$.

For example, if we had a set in the form,

$$A_1=left{z(x,y,z):x,y,zinmathbb{Z}right}cap(0,1)$$

Which is a subset of $(a,b)=(0,1)$, where $z:x,y,zto mathbb{R}$; how would we partition $(a,b)$ into $m$ equal sub-intervals and count the number of sub-intervals that intersect with this form of $A_1$?

Would the code be fast enough for large $m$?

How about other general forms?

Given a line segment $PQ$ and a straight line $L$ not intersecting….

Given a line segment $PQ$ and a straight line $L$ not intersecting the line segment. At what point on $L$ will $PQ$ subtend the greatest angle.

I am looking for a geometric result..

Thanks.

geometry – Intersecting triangles: Find the length of a segment given one side and two ratios

Diagram described in the problem

I’m trying to find the length of the segment from H to J. The problem states the length of the segment from B to E is 494, along with a couple ratios of the sides. BC:CD = 2:3, and DE:EF:FG = 2:3:4.
The problem also makes it clear that B, C, and D are collinear; D, E, F, and G are collinear; C, H, and G are collinear; B, H, J, and E are collinear; and C, J, and F are collinear, as seen in the diagram.

Since they give the length of BE, I believe I am supposed to find the ratio between BE and HJ. However, they don’t give any angles, nor relate the two given ratios together, so I’m doubting that there are any similar triangles or angle bisectors that could be used.

I remember reading that Ceva’s theorem can be used to relate the ratios of sides, but upon further research, there should be three cevians and a point which they meet for the theorem to be used. Could some of the lines be extended to achieve this effect? Or is this even the right approach to the problem?

linear algebra: a problem with intersecting lines

if I save some line equations (y = ax + b) as follows for each line, with two points:
(0, b), (1, a + b), and none of them intersect with another, for example:

$ y = x + 1 -> (0.2), (1,2) $

$ y = x + 2 -> (0,2), (1,3) $

$ y = x + 3 -> (0,3), (1,4) $

and I want to add another line, only if it doesn't intersect my previous lines between $ 0 <= x <= 1 $
for example i want to add y = 6x-1 ((0, -1), (1,5))
Why is it enough to check if it intersects the predecessor / successor on the y value?
for example, here it is enough to verify it with y = x + 1

Circular arcs not intersecting from the origin

I am looking for a general way to find the midpoint and radius of a circle that intersects an origin and an arbitrary point so that the arc between the points does not intersect with any other arc of the same origin. I am not even sure if this problem can be solved.

at.algebraic topology – No additivity of intersecting forms

Two oriented dice $ 4k $-collectors $ X_1 $ Y $ X_2 $, Novikov's additivity tells us that
$$
sigma (X_1 sharp X_2) = sigma (X_1) + sigma (X_2). $$

More generally, if we paste the limits of two such varieties $ X_1 $ Y $ X_2 $ Through a diffeomorphism that reverses the orientation (or even only part of the limits, see here), Novikov's additivity is maintained. Wall has analyzed the failure of additivity when we paste along a sub-file of the border which in turn has a limit.

I am interested in how this story extends to the intersection forms. In particular:

What can be said about the relationship between the form of intersection? $ Q_ {X} $, where $ X = X_1 cup_ phi X_2 $ where $ phi $ It is a diffeomorphism that reverses the orientation between closed submanifolds of the limit and the forms of intersection. $ Q_ {X_1} $ Y $ Q_ {X_2} $?

I know that the additivity of the intersection forms is maintained in the case where the limits of $ X_1 $ Y $ X_2 $ they are two homology spheres and are pasted along their entire border, in particular, $ Q_ {X_1 sharp X_2} = Q_ {X_1} + Q_ {X_2} $. I wonder if there is any error term in the general case that has been identified. For what it's worth, I'm interested mainly in the case where $ k = 1 $.