ag.algebraic geometry – Calculation with ideals / twisted cubic

i have some problems with the calculations in the ideals. I recently studied algebraic geometry and the concept of projective closure. There is a counterexample which has just homogenized the base of an ideal, does not give the projective closure of the twisted cube. You will find it under: Projective closure of the affine curve

I don't see why the polynomial $ p_1 = y ^ 2-xz $ is an element of the ideal $ (x ^ 2-y, x ^ 3-z) $. I know that the ideal is generated in a finite way. So I have to solve the following equation $ p_1 = r_1 * (x ^ 2-y) + r_2 * (x ^ 3-z) $ for $ r_1, r_2 in k (x, y, z) $, which I find quite difficult. Are there easy arithmetic rules for ideals?

This brings me to a more general reflection. For example: $ frac {k (x, y, z)} {(z-1, x ^ 2-y)} = k (x, x ^ 2) $ there you use it $ z = $ 1 and $ y = x ^ 2 $. In the same sense, you can show that $ (x ^ 3-z, x ^ 2-y, xy-z) $ is generated by only two elements:

$ (- 1) (x ^ 2-y) * x = (x ^ 3-y * x) (- 1) = (z-xy) (- 1) $

last equation is valid because $ x ^ 3 = z $. (I saw it in a conference)

Can i use it for problem? If this is true, why is it mathematically correct?

arithmetic geometry – How does an analytic space correspond to a Banach $ p $ -adic space

Let $ K $ to be a finite extension of $ mathbb {Q} _p $, and $ V $ to be a Banach algebra on $ K $, then what is the $ K $-analytical space corresponding to $ V $? What is the definition of $ K $-analytical space? This is mentioned in the first paragraph on page 6 of the article by Laurent Berger.

Likewise, if $ W $ is a Banach algebra on $ mathbb {C} $, so $ W $ corresponds to a complex analytical space? I only know for a variety $ mathbb {C} $, we can do it and we have GAGA.

Thank you!

discrete geometry – On the triangulation of the plane using edges of rational length

Basic question: Can the Euclidean plane be divided into a vertex-to-vertex arrangement of non-overlapping triangles so that each edge has a unique rational length which is between 1 and a specific rational R greater than 1?

If possible, additional constraints can be applied such as "all triangles must have an equal area (OR an equal perimeter)". Alternatively, one can relax the requirement from top to top. We can also replace the requirement that each edge has a unique length with (say) the triangles being out of line two by two.

Note: Requiring that the lengths of all edges be integers rather than rationals would result in the lengths of the edges being unlimited even if triangulation with all edges having unique lengths is possible (I don't know if that is possible).

analytical geometry – polygon lines shifted in the opposite direction of the polygon (inflation)

Let's say that a convex polygon is given as a set of coordinates. We have to shift all the lines (to which the edges belong) by $ h $ in the direction of their perpendicular lines and in the opposite direction of the polygon.

More simply, the polygon should be somehow inflated. If one of the given lines satisfies the equation $ y = kx + b_1 $ then we just need to find the coordinates $ b_2 $ for the offset line using the formula $ | b_2-b_1 | = h sqrt {k ^ 2 + 1} $.

I don't know how to choose the right direction. I know we can check the orientation of a point in relation to a line but in this case I have to make sure that the new line has moved away from all the points of the polygon .

Thanks in advance!

dg.differential geometry – Operator Index Fredholm

I have two vector bundles $ E_1 $, $ E_2 $ more than $ M $ and integration of smooth sections $ lambda: Gamma (M, E_1) rightarrow Gamma (M, E_1 oplus E_2) $. I consider a Fredholm differential operator $ D_1: Gamma (M, E_1) rightarrow Gamma (M, E_1) $ which can be easily lifted to the differential operator Fredholm $ D_2: Gamma (M, E_1 oplus E_2) rightarrow Gamma (M, E_1 oplus E_2) $. What to say about their clues $ Ind (D_1) $ and $ Ind (D_2) $?

ag.algebraic geometry – Global functions on a product of diagrams on the artinian ring

For a morphism of patterns $ f: X to S $ with $ S = text {Spec} (R) $ refine let's write $ A (X) = H ^ 0 (X, mathcal {O} _X) $. I'm interested in the morphism of $ R $-algebras
$$
c: A (X) otimes_R A (Y) to A (X times_SY)
$$

for some people $ X, Y $ more than $ S $ that I don't want to assume refined or appropriate. Without condition of affinity, the classic case where $ c $ turns out to be an isomorphism is when $ X to S $ is flat and $ A (Y) $ is flat (SGA3 I, Lemma 11.1).

I want to prove that $ c $ is an isomorphism in the following case: $ R $ is an artinian ring with an algebraically closed residue field and $ X $ and $ Y $ are smooth (connected if it helps) on $ R $.

In fact, in my intended application, I have $ X = Y = G $ a smoothly connected group scheme, which by Chevalley we know is an extension of an abelian variety by an affine group scheme, or by the "double" Chevalley is an extension of an affine by an anti -refined. But I'm not sure it is relevant in any way.

My attempts to prove it so far were as follows. Yes $ X, Y $ are both affine, or both appropriate, so the statement is true. I tried unscrewing, that is to say $ X to S $ factors like an affine morphism $ X to Z $ followed by good body shape $ Z to S $ (for plans, Temkin 2011; for group plans, it's Chevalley) but couldn't really go any further.

Another approach that I tried was to use Künneth's formula (let's say in the form
stacks up the Tag 0FLQ project. taking $ H ^ 0 $ in Künneth's formula, in the RHS, I get exactly $ A (X times_SY) $, while in the LHS algebra $ A (X) otimes_R A (Y) $ should appear as a graduated piece of the stop for the second spectral sequence of hypercohomology, if I'm right. The problem here is that I don't know how to write this spectral sequence, because instead of a functor, in this case, I have a bifunction (i.e. $ otimes $) and then I get lost.

Note that even though I would be quite surprised, it's still possible that my assumption for isomorphism is wrong, and I would be happy (well … would I do it?) To be rebutted.

Thanks for any help or suggestion!

ag.algebraic geometry – Identify line bundles on $ mathbb {P} ^ n $

Consider the complex projective space $ mathbb {P} ^ n $. It corresponds to a principal $ mathbb {C} ^ * $-package
$$ mathbb {C} ^ * to mathbb {C} ^ {n + 1} -0 to mathbb {P} ^ n, $$
or $ mathbb {C} ^ * $ acts on $ mathbb {C} ^ {n + 1} -0 $ by multiplication
$$ t cdot (z_0, dots, z_n) = (tz_0, dots, tz_n). $$
Now fix an integer $ d $. We can change the fiber $ mathbb {C} ^ * $ at $ mathbb {C} $ considering the action of $ mathbb {C} ^ * $ sure $ mathbb {C} $,
$$ t cdot z = t ^ d z. $$

Then we get a bundle of lines $ left ( mathbb {C} ^ {n + 1} -0 right) times _ { mathbb {C} ^ *} mathbb {C} $ more than $ mathbb {P} ^ n $, or $ mathbb {C} ^ * $ acts on $ left ( mathbb {C} ^ {n + 1} -0 right) times mathbb {C} $ through
$$ t cdot ((z_0, dots, z_n), z) = ((tz_0, dots, tz_n), t ^ d z). $$

Is the group of lines $ left ( mathbb {C} ^ {n + 1} -0 right) times _ { mathbb {C} ^ *} mathbb {C} cong mathcal {O} (d)? $

The following is my attempt to identify $ left ( mathbb {C} ^ {n + 1} -0 right) times _ { mathbb {C} ^ *} mathbb {C} $ with $ mathcal {O} (d) $. I use the standard open lid $ {U_i } _ {i = 0, points, n} $ of $ mathbb {P} ^ n $ (i.e., $ x_i neq 0 $ sure $ U_i $) and trivialize the bundle of lines on each $ U_i $ as following:
$$
begin {aligned}
left ( left ( mathbb {C} ^ {n + 1} -0 right) times _ { mathbb {C} ^ *} mathbb {C} right) | _ {Ui} & to U_i times mathbb {C} \
((z_0, dots, z_n), z) & mapsto ((z_0, dots, z_n), z_i ^ {- d} z).
end {aligned}
$$

This is well defined because another representative $ ((tz_0, dots, tz_n), t ^ d z) $ matches the same item.

The transition function of $ U_i $ at $ U_j $ is then
$$
begin {aligned}
g_ {ji}: U_i cap U_j & to mathbb {C} ^ * \
(z_0, dots, z_n) & mapsto (z_j / z_i) ^ {- d}.
end {aligned}
$$

So these $ g_ {ji} $are transition functions for $ mathcal {O} (d) $.

Does all of the above seem correct?

calculation geometry – Robust calculation of intersection points of two 2D segments / lines

Given two line segments, the problem is to find a point of intersection of the corresponding lines (assuming that they are not parallel or coincident).

There is a Wikipedia article that gives us exact formulas, but there are two: the one that uses t report and approach the intersection point of the first line segment and the other – uses u and the second line segment. How can I select the one to use in my scenario?

For example: my initial implementation which has always used t has failed

first_segment = Segment(start=Point(x=-5, y=0), end=Point(x=72057594037954921, y=0))
second_segment = Segment(start=Point(x=0, y=0), end=Point(x=0, y=3))

given

Point(x=5.921189464665284e-16, y=0.0)

which is incorrect but when i switch to using u or change the order of arguments, this gives me correct

Point(x=0.0, y=0.0)

So my question is this: is there a robust way to calculate the point of intersection?

complex geometry – Non-standard definitions of complexifications

I started studying Daniel Huybrechts' book, Complex Geometry An Introduction. I tried to study as much back as possible, but got stuck on the concepts of almost complex structures and complexification. I have studied several books and articles on the subject, notably those of Keith Conrad, Jordan Bell, Gregory W. Moore, Steven Roman, Suetin, Kostrikin and Mainin, Gauthier

I have several questions on the concepts of almost complex structures and complexification. Here is some:

I notice that the standard definitions of the complexification of a $ mathbb R- $ the vector space is as follows:

  1. In terms of direct sums, $ V ^ { mathbb C, sum}: = (V ^ 2, J) $ or $ J $ is the almost complex structure $ J: V ^ 2 to V ^ 2, J (v, w): = (- w, v) $ which corresponds to the complex structure $ s _ {(J, V ^ 2)}: mathbb C times V ^ 2 to V ^ 2, $$ s _ {(J, V ^ 2)} (a + bi, (v, w) ) $ $: = s_ {V ^ 2} (a, (v, w)) + s_ {V ^ 2} (b, J (v, w)) $$ = a (v, w) + bJ (v , w) $ or $ s_ {V ^ 2} $ is true scalar multiplication over $ V ^ 2 $ extended to $ s _ {(J, V ^ 2)} $. In particular, $ i (v, w) = (- w, v) $

  2. In terms of tensor products $ V ^ { mathbb C, tensor}: = V bigotimes mathbb C $. Here, $ mathbb C $ scalar multiplication is as follows on elementary tensors $ z (v otimes alpha): = v otimes (z alpha) $, for $ v in V $ and $ z, alpha in mathbb C $.

I notice that we can have a different definition of the sum $ V ^ { mathbb C, sum, -J}: = (V ^ 2, -J) $, or $ mathbb C $ scalar multiplication is now $ i (v, w) = (- J) (v, w): = -J (v, w): = (w, -v) $.

  • Note: In this notation, $ V ^ { mathbb C, sum, J} = V ^ { mathbb C, sum} $.

Question 1: $ V ^ { mathbb C, sum, -J} $ somehow correspond to $ V ^ { mathbb C, tensor, f (z) = overline z}: = (V bigotimes mathbb C, f (z) = overline z) $, or $ mathbb C $ scalar multiplication is as follows on elementary tensors $ z (v otimes alpha): = v otimes (f (z) alpha) $ $ = v otimes ( overline z alpha) $, for $ v in V $ and $ z, alpha in mathbb C $?

  • Note: In this notation, $ V ^ { mathbb C, tensor, f (z) = id _ { mathbb C} (z)} = (V bigotimes mathbb C) $

    • Note: any general correspondence between almost complex structures $ K $ sure $ V ^ 2 $ and the $ f $is on $ V bigotimes mathbb C $ can be reserved for question 2. For question 1, I'm interested to see if $ V ^ { mathbb C, sum, -J} $ and $ V ^ { mathbb C, tensor, f (z) = overline z} $ are "more isomorphic" than $ V ^ { mathbb C, sum, -J} $ and $ V ^ { mathbb C, sum, J} $ (I think Gauthier looks like they are not $ mathbb C $-isomorphic by identity card or something) in the sense that $ V ^ { mathbb C, sum, -J} $ and $ V ^ { mathbb C, tensor, f (z) = overline z} $ are not only $ mathbb C $-isomorphic, but $ mathbb C $-isomorph is a unique way, I guess, like Keith Conrad's Theorem 3.1 or here

Question 2: What is the correspondence with almost complex structures $ K $ sure $ V ^ 2 $ and the $ f $is on $ V bigotimes mathbb C $?

algebraic geometry – Questions regarding Gathmann's motivations for schemas

Let $ X $ be an algebraic set, i.e. a closed subset of $ mathbb {A} ^ n $. so $ X $ can be written as a finite union of irreducible subsets, its irreducible components. Generalize affine varieties as locally ringed spaces and define varieties as ringed spaces

  1. are irreducible / connected
  2. admit a finite open cover of affine varieties
  3. have a sheaf of $ k $-functions evaluated (closed alg)

can we say that prevalence co-products generalize (intrinsically) the notion of an algebraic whole? If so, can we accomplish the same thing by removing condition 1. above?

I ask because Gathmann says the following as one of the motivations of diets

5.1. Affine diagrams. We now come to the definition of diagrams, which are the main objects of study in algebraic geometry. The notion of schemas extends that of precariousness in several ways. We have already encountered several cases where an extension of the precariousness category could be useful:

• We defined a prevalence as irreducible. Obviously, it makes sense to also consider reducible spaces. In the case of affine and projective varieties, we have called them algebraic sets, but we have given them no other structure nor defined regular functions and morphisms. Now we want to make collapsible spaces complete objects of our category.

A second (third?) Question concerns the motivation for monitoring:

At the present time, we have no geometric objects corresponding to non-radical ideals in $ k (x_1, ldots, x_n) $, or in other words to coordinate rings with nilpotent elements. However, these non-radical ideals naturally arise: e. g. we saw in exercise 1.4.1 that the intersections of the affine varieties correspond to the sums of their ideals, modulo taking the radical. It would seem more natural to define the intersection $ X_1 cap X_2 $ of two affine varieties $ X_1, X2 subset mathbb {A} ^ n $ to be a geometric object associated with the ideal $ I (X_1) + I (X_2) ⊂ k (x_1, ldots, x_n) $.

Could we not simply "forget" the Nullstellensatz and find a non-radical theory in the varieties? However, we move without Nullstellensatz for the diets, could we not do the same with the varieties?