Aggelematic geometry – Characterization of rational polyhedral cone faces

This is a basic (seemingly) basic lemma on rational polyhedral cones that is sometimes used to work with toric varieties and is generally "left to the reader". Unfortunately, I could neither prove it myself nor find complete proof in the literature. So I'm looking for either proof or an appropriate reference.

Lemma. Let $$V$$ bean $$mathbb {R}$$-vector of finite dimensional space, and let $$N$$ be a $$mathbb {Z}$$-structure on $$V$$ (that is, a free abelian group with $$N otimes _ { mathbb {Z}} mathbb {R} = V$$). Let $$sigma$$ and $$tau$$ be $$N$$rational polyhedral cones in $$V$$and suppose that $$tau$$ is a subset of $$sigma$$. Then, the following statements are equivalent:

(I) $$tau$$ is a face of $$sigma$$;

(ii) if $$x, y in sigma$$ with $$x + y in tau$$then $$x, y in tau$$;

(iii) if $$x, y in sigma cap N$$ with $$x + y in tau$$then $$x, y in tau$$.

Show that (i) and (ii) are equivalent and that (iii) is clear. My problem is to show that (iii) implies (i) or (ii), that is to say, it suffices to consider only the rational points.

Note 1 Try to show (iii)$$Rightarrow$$(i) in a manner similar to (ii)$$Rightarrow$$(i) asks whether $$( sigma- tau) cap N = ( sigma cap N) – ( tau cap N)$$, which has a negative answer in general.

Note 2 Condition (iii) is sometimes expressed by saying that the monoid $$tau cap N$$ is a monoid face $$sigma cap N$$and likewise for (ii).

Arithmetic Geometry – Weighted Projective Lines and Elliptic Curves

Low-level modular curves can sometimes be described as weighted projective lines. For example, on $$mathbb {Z} (1/2)$$ the compacted pile of elliptic curves with a complete level 2 structure is isomorphic to the projective degree line $$(2, 2)$$. More than $$mathbb {Z} (1/3)$$ the compact pile of elliptic curves with $$Gamma_1 (3)$$ the level structure is the isomorphism of the projective line of the degree $$(1, 3)$$ (as mentioned by Meier in examples 2.1 https://arxiv.org/abs/1710.03461).

Is there an explicit description of the compacted stack of elliptic curves with a complete Level 3 structure noted somewhere?

matrices – Plucker line coordinate convention in multi-view geometry?

In Hartley and ZisserMan's book "Multiple View Geometry",
The plucker coordinates have been presented as cells of the asymmetric matrix which is the homogeneous two-point wedge product:

with the definition for coordinates:

I still need to read the section on the use of these on the projected lines of the same book, but for now, I've found this strange convention compared to the convention of {direction, moment} which is common to see;
In this Hartley / ZisserMan notation, it seems that the direction vector is
(-14, l42, -34) and the moment vector (p (2,3), – p (1,3), p (1,2)).
I wondered if this notation was a special notation for any reason.

Thank you!

Geometry – Long projection so as not to distort the distance

I have a lot of circles that are long positions and a radius in meters.

I need to check if two circles intersect and I can use this haversine distance function.

However, to speed up performance, I would like to build a quadrilateral tree containing all the circles.

From what I understand, I have to project the last positions in a projection without distance distortion.

This would allow me to a) check the intersection of the circle using a normal dist function, but especially b) allow me to check the intersection of the circle and the rectangle to build a quadrilateral tree.

I have been searching for centuries and I have not found a way to make such a projection?

I understand that such a projection would result in an oval space (I think) instead of a rectangular space and that, therefore, the implementation of a quadtree would not be perfect, but sufficient to improve performance.

Also, if you know of another way to create a tree with four branches in these circles, please let me know.

Differential Geometry – How Recessed and Isothermal Transforms Change Curvature?

Let $$X$$ be a (closed) surface (or in general a variety). Let $$g$$ to be a Riemannian metric on it. I'm thinking about how the following operations modify the curvature of $$g$$:

(1) withdrawal by some diffeomorphism of $$X$$;

(2) multiplication by a scalar, that is to say $$lambda g$$ for some people $$lambda in mathbb R$$.

More specifically, can we use these two operations to make it constant?

Ag.algebraic geometry – Are the coherent smooth sheaves of rank 1 always the same as the bundles of lines?

assume $$mathcal {L}$$ more than $$mathbb {CP} ^ 2$$ is given by the following short exact sequence,

$$0 rightarrow mathcal {L} rightarrow mathcal {O} (4) oplus mathcal {O} (2) oplus mathcal {O} (8) oplus mathcal {O} (1) oplus mathcal {O} (1) rightarrow mathcal {O} (14) oplus mathcal {O} (8) oplus mathcal {O} (5) oplus mathcal {O} (2) rightarrow 0.$$

It's easy to verify that $$mathcal {E} xt ^ i ( mathcal {L}, mathcal {O}) = 0$$ for $$i ge1$$. So $$mathcal {L}$$ is a coherent sheaf of rank 1 smooth. So I'm waiting $$mathcal {L}$$ should be a line pack but $$Ch_2 ( mathcal {L}) ne frac {1} {2} c_1 ( mathcal {L})$$, contrary to what happens for the bundles of lines!

I do not know what's wrong here …

stacks – Prerequisites for understanding the algebraic geometry of "algebraic sheaves"

I'm trying to learn about the algebraic geometry of Gerbes.

I am familiar with the configuration of the sheaves in the case of differential geometry. Although there is some similarity between the differentiable sheaves and sheaves mentioned above, they are not quite the same.

So, what are the prerequisites for learning sheaves in the configuration of differential geometry? I am familiar with the concept of batteries. What other Grothendieck topologies and overlays on categories with these Grothendieck topologies should I be comfortable with to understand and use the sheaf concept mentioned above.

Aggraphic geometry – Multiplicity of a positive characteristic polynomial

Let $$mathbb K$$ to be an area of ​​character $$p> 0$$.
Let $$f in mathbb K [x_1, dots, x_n]$$ to be a multivariate polynomial and let $$q in mathbb K ^ n$$. Is there a computer method for determining the multiplicity of $$f$$ at $$q$$ without explicitly calculating with the Gröbner bases the powers of the determining ideal of $$q$$?

Geometry: what is the angle $phi$, if the area of ​​the yellow rectangle is equal to the area of ​​the red triangle?

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Aggressive Geometry – Do regular (but not smooth) conics on a value field distinct from the $2$ feature admit a regular pattern on the valuation ring?

Let $$K$$ to be a field of character not perfect $$2$$. Let $$T subseteq K$$ to be a discreet evaluation ring.
Assume that there is $$a, b in K ^ { times}$$ such as the projective conic $$C$$ Defined by $$aX ^ 2 + b Y ^ 2 + Z ^ 2 = 0$$ does not contain $$K$$rationalization point
(in particular $$a, b$$ and $$ab$$ are not squares $$K$$). then $$C$$ is not smooth on $$K$$ at each point. however, $$C$$ is regular (see exercise 4.3.22d) of Qing Liu's book Algebraic geometry and arithmetic curves).

Question: Is $$C$$ have a regular projective model on $$T$$ (that is, a regular fiber-optic projective surface on $$T$$ with generic fiber isomorphic to $$C$$)?

Each smooth curve on $$K$$ admits a regular projective model on $$T$$, as indicated (for example) in Corollary 8.3.51 of Liu's aforementioned book.
This seems to be based on Lipman's resolution of the singularities of $$2$$-dimensional excellent Noetherian patterns, applied to a normal pattern of the curve on $$T$$, or rather a basic change at the end of $$T$$ (Excuse the sloppy description, it may not be a very accurate description of what is really happening).
In any case, it seems crucial to use the fact that the generic fiber of the normal model is not only regular, but smooth $$K$$.

Nevertheless, this is a general assertion and it is a priori possible that, in concrete cases of regular projective curves (such as all integral and regular conics $$K$$, which interests me), there are regular projective models on $$T$$, whether the regular curve is smooth or not $$K$$ or not.