algebraic geometry – Plane minus a projective line vs. plane minus a conic

Let $k$ be a field. In $mathbb{P} = mathbb{P}^2(k)$, an irreducible conic $C$ is isomorphic to a projective line $U = mathbb{P}^1(k)$ (which we take to be a line in $mathbb{P}$). If I am not mistaken, we have that $mathbb{P} setminus U$ is birational to $mathbb{P} setminus C$. What is the easiest way to see this ?

mg.metric geometry – tetrahedral interpolation and integration along a segment

Let’s say we have a several tetrahedrons $T_i$ whose faces touch so that each face belong to two tetrahedrons. Each tetrahedron contain a value $V_{i}$.

Given a position $P$ inside the tetrahedron $T_0$, and neighboring tetrahedron are labeled $T_1, T_2, T_3, T_4$.

How to compute the value $V(P)$ such that its value is a linear interpolation between all $V_i$?

Following this, given a direction $vec{d}$ and the origin $O$ and a scalar $t$ such that $P(t)=O+d*t$, what is the equation giving the interpolated value along this segment $V(t)$, considering only the part where the segment is inside $T_0$?

I tried to use barycentric coordinates, and I think it confused me more than it helped.

What would be a simple explanation for solving such a problem?

ag.algebraic geometry – Finitely generated commutative rings with the same profinite completion

Let $R_1$ and $R_2$ be two finitely generated commutative rings. Assume that their profinite completions are isomorphic: $widehat{R_1}cong widehat{R_2}$.

Suppose that $R_1$ is a domain. Does it imply that $R_2$ is a domain as well?

The isomorphism between $widehat{R_1}$ and $widehat{R_2}$ is equivalent to the existence of a bijection $phi: operatorname{maxSpec}(R_1)to operatorname{maxSpec}(R_2)$ between the sets of maximal ideals, such that for all $Min operatorname{maxSpec}(R_1)$ the corresponding completions $(R_1)_{(M)}$ and $(R_2)_{(phi(M))}$ are isomorphic.

ag.algebraic geometry – Direct proof that $mathbb{P}^n$ is rationally connected

Let $mathbb{P}^n$ denote projective $n$-space over $mathbb{C}$. I am almost certain that this question is well-known to algebraic geometers, but I cannot find a reference.

Q: Is there a direct proof of the fact that $mathbb{P}^n$ is rationally connected?

Of course, a result of Kóllar-Miyaoka-Mori tells us that something more general is true, namely, that any Fano variety is rationally connected. The proof seems quite long, however, so I am naturally interested in whether a simpler, more direct, proof can be given for $mathbb{P}^n$.

reference request – Which books should I read in order to be prepared to study Information Geometry?

At the moment, I am preparing my master’s thesis (in statistics) and I intend to keep studying in order to pursue a doctoral degree. To be precise, I am mainly interested in studying Information Geometry.

It is worth emphasizing that I have a bachelor’s degree in pure mathematics.

Having said that, I would like to be advised as to which books should I study in order to prepare myself to get acquainted to this subject.

More precisely, could someone tell me which books of differential geometry, probability and statistics should I read in order to introduce myself to it?

A progressive list of readings would be appreciated.

ag.algebraic geometry – Relation between subspaces of diagonal matrix and its “sign” matrix

Let $D$ be a $n times n$ diagonal matrix with both positive and negative (but all non-zero) entries. Let $J = sign(D)$ be the matrix of $1$s and $-1$s representing the signs of the entries of $D$. Suppose there are two orthogonal subspaces of $mathbb{R}^n$, with bases given by the columns of matrices $X$ and $Y$ respectively, that together span $mathbb{R}^n$. In other words $X^T Y = 0$, and $begin{bmatrix} X & Y end{bmatrix}$ is an $n times n$ full rank matrix (the number of columns of $X$ and $Y$ don’t have to be equal).

If we are given that $D$ is positive definite on $X$ and negative definite on $Y$, can we say anything about the definiteness of the sign matrix $J$ on $X$ and $Y$? More concretely, is the following statement true?

$X^T D X succ 0$ and $Y^T D Y prec 0 implies$ $X^T J X succ 0$ and $Y^T J Y prec 0$.

I can prove that this is true for $n = 2$ (i.e. when $X$ and $Y$ are both $2 times 1$). But I have no idea how to prove that this is true in general, or even an intuition for why it should be. But I’ve performed enough simulations to convince myself that it is true. Note that both conditions on $X$ and $Y$ (orthogonality and spanning of $mathbb{R}^n$) are needed for this result to hold.

Any help here would be highly appreciated!

geometry – Cevians and areas

In an ABC triangle, the cevianas AD, BE and CF compete in P. Show that

$frac{S_{DEF}}{2S_{ABC}}=frac{PD .PE.PF}{PA.PB.PC}$
Using areas relation, I found

$frac{3S_{ABC}-overbrace{(S_{PAB}+S_{PAC}+S_{PBC})}^{S_{ABC}}}{S_{ABC}}=frac{PA}{PD}+frac{PB}{PE}+frac{PC}{PF}implies
2S_{ABC}=S_{ABC}left(frac{PA}{PD}+frac{PB}{PE}+frac{PC}{PF}right)\
frac{S_{PDF}}{S_{PAC}}=frac{PD×PF}{PA×PC}\
frac{S_{PDF}}{S_{PAB}}=frac{PD×PE}{PA×PB}\
frac{S_{PEF}}{S_{PBC}}=frac{PE×PF}{PB×PC}\
S_{PDF}+S_{PED}+S_{PFE}=S_{DEF}$

Can someone help me to finish this proof?
Thanks for antetion!

postgresql – How to get the type of geometry column in postgres?

I need to determine the type of geometry column and doing it with the following query:

SELECT type 
FROM   geometry_columns 
WHERE  f_table_schema = 'public' 
       AND f_table_name = 'table_name' 
       AND f_geometry_column = 'col_name' 

It works fine for regular geometries, like Point or Polygon. But then there are also some fancy geometries like PointZ. How do I determine if column has it? Above mentioned query returns just POINT for it.

ag.algebraic geometry – On the intersection numbers of the generators of $text{Pic}(X)$ of a smooth quintic surface

There exists the following result in the literature: There exists a polarized $K3$ surface $(X, H)$ of genus $3$ and a smooth irreducible curve $C$ on $X$ satisfying $C^2 =4$, $C.H=6$ such that $text{Pic}(X) cong mathbb Z(H) oplus Z(C)$. The theorem follows from (https://arxiv.org/pdf/math/9805140.pdf) theorem $1.1(iv)$.

Now Let’s consider $X$ to be a smooth quintic hypersurface in $mathbb P^3$ with Picard number $2$. Then I think it can be shown that $text{Pic}(X) cong mathbb Z(H) oplus Z(H’)$, where $H$ is the hyperplane class and $H’$ is some divisor. Now in order to locate the ample line bundles in $text{Pic}(X)$ using the Nakai-Moishezon criterion, we must know the intersection numbers $H.H’$ and $H’^2$.

In this context my question is the following: Does there exist in the literature an analogous existence result as the first-mentioned theorem for smooth quintic hypersurface with Picard number $2$?

To be more precise: Does there exist a polarized smooth quintic hypersurface $(X, H)$ in $mathbb P^3$ and a smooth irreducible curve $C$ for which the intersection numbers $C^2$ and $C.H$ are known and $text{Pic}(X) cong mathbb Z(H) oplus Z(C)$?

Can someone give me any reference which could be even remotely useful in the context of finding out such $(X,H)$ and $C$

Any help from anyone is welcome.

ag.algebraic geometry – Characterization of injective sheaves

Let $mathcal{C}$ be a site with enough points, and let $R$ be a noetherian ring. Is it true that a sheaf $F$ of $R$-modules on $mathcal{C}$ is injective if and only if for every morphism $Uto V$ in $mathcal{C}$ the induced homomorphism $F(V)to F(U)$ is a split surjective homomorphism of injective modules?

I know this if $mathcal{C}$ is the poset of open subsets of a topological space, but the proof does not seem to generalize: it uses the fact that, if $A$ is a subsheaf of the constant sheaf $R$, the locus of points with stalks equal to $R$ is open, and I don’t know how to make sense of this under my more general assumptions.