Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [projective-geometry]

The tag has no usage guidance.

13
votes
1answer
289 views

Projective-invariant differential operator

This question was originally asked on Math StackExchange. Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that \begin{align*} &T(g) = ...
-1
votes
0answers
179 views

On an exercise in section 4 of Chapter I from Hartshorne's Algebraic Geometry

It is about exercise 4.9: Let $X$ be a projective variety of dimension $r$ in $\mathbb{P}^n$ with $n\geq r+2$. Show that for suitable choice of $P \notin X$ and a linear $\mathbb{P}^{n-1}\subseteq \...
2
votes
0answers
126 views

A problem of four conics

I found a remarkable theorem of four conics as follows some years ago. But it has no proof; I am looking for a proof: Theorem: Take three conics. Suppose that each of them touch a fourth conic at two ...
4
votes
1answer
248 views

Is it a new method to construction of a conic, how can prove?

There are some methods to construct a conic, example: Based on Pascal theorem, Steiner construction, .....I propose a method to construct a conic as follows: Let $L_1, L_2$ be two parallel lines, ...
2
votes
1answer
145 views

Linear subspaces in quadric hypersurfaces

Consider $H_1,H_2,H_3\subset\mathbb{P}^{2m+1}$ three general linear subspaces of projective dimension $m$. Then there exists a quadric hypersurface $Q^{2m}\subset\mathbb{P}^{2m+1}$ containing $H_1,...
2
votes
0answers
61 views

Projectively flat Weyl connection on closed higher genus surface

A Weyl connection on a smooth $n$-manifold $M$ is a torsion-free connection $\nabla$ on its tangent bundle that preserves some conformal structure $[g]$ on $M$. By this I mean that its parallel ...
2
votes
0answers
145 views

Moduli space is a Calabi-Yau manifold?

I asked a question here where a moduli space of flat connection is related to the $n$-dimensional complex projective space: $$\Bbb E/S_n \cong \Bbb P^{n-1}. $$ This is related to a 4d SU(N) Yang-...
13
votes
2answers
491 views

Seeking a more symmetric realization of a configuration of 10 planes, 25 lines and 15 points in projective space

I've got ten (projective) planes in projective 3-space: \begin{align} &x=0\\ &z=0\\ &t=0\\ &x+y=0\\ &x-y=0\\ &z+t=0\\ &x-y-z=0\\ &x+y+z=0\\ &x-y+t=0\\ &x+y-t=0 ...
2
votes
0answers
38 views

Is the finite projective plane stable as an extremal set system?

Let $\Sigma$ be a set of $|\Sigma| = n$ subsets of the universe $[n]$, each of size $k$, with the property that any two of these subsets intersect on at most one element. It is easy to see that the ...
3
votes
0answers
72 views

Are there local invariants for smooth planes?

A smooth plane is a smooth double fibration $$ \mathbb{RP}^2 \overset{\pi_1}{\longleftarrow} PT\mathbb{RP}^2 \overset{\pi_2}{\longrightarrow} \mathbb{RP}^2 $$ where the system of curves $\pi_1(\pi_2^{...
3
votes
1answer
84 views

Osculating spaces of intersection of two varieties

Let $Z = X\cap Y\subset\mathbb{C}^N$ be a manifold given as the intersection of two manifolds $X,Y$ intersecting transversally along $Z$. Let $T_p^kX,T_p^kY,T_p^kZ$ be the $k$-osculating spaces at $p\...
2
votes
0answers
62 views

Boundary at infinity for projective manifolds?

Given a compact real projective manifold, is there something "like" a boundary at infinity for its universal cover? In the case of compact projective manifolds obtained from divisible convex sets (...
22
votes
6answers
1k views

About the definition of E8, and Rosenfeld's “Geometry of Lie groups”

I've been searching the literature for a direct definition of the group $E_8$ (over a general field, but even a definition of just one incarnation would be great). I knew (from talking to people) that ...
5
votes
1answer
205 views

Intersections in $\mathbb{P}^1\times\mathbb{P}^1$

Let $F$ be an algebraically closed field and $\mathbb{P}^1$ the projective line over $F$. Suppose $V_1, V_2$ are two 1-dimensional subvarieties of the 2-dimensional variety $\mathbb{P}^1\times\mathbb{...
0
votes
0answers
108 views

Derived Category of the Fano 4fold variety of lines

Let $X\subset P^5$ be a smooth cubic fourfold. It is well known that its variety of lines $F(X)$ is a smooth fourfold Fano variety. Hence its derived category should have a semi-orthogonal ...
1
vote
0answers
170 views

Dimension projectivised tangent space equal dimension variety+1

I'm reading some lecture notes (unfortunatey in italian, https://me.unitn.it/system/files/Bernardi%20Alessandra/tesi_teroni_1.pdf , page 5), where there's this statement (without proof): Suppose we ...
6
votes
1answer
258 views

Generalized projective spaces, spheres, and exotic spheres [closed]

I like to explore and ask for proper references for the relations between generalized projective spaces, spheres, and exotic spheres: The real projective space $\mathbb{RP}^1 \simeq S^1,$ is ...
5
votes
0answers
211 views

$N$-$th$ closed chain of six circles

Since 2013, I found a very nice configuration: $N$-th closed chain of six circles. This is a generalization of theorem 1, problem 2 in here and theorem 2 in here and here (and is also generalization ...
2
votes
1answer
181 views

Yiu's equilateral triangle-triplet points

In more than 2300 years since Euclid's Elements appear, there were only two equilateral triangles become famous: The Morely equilateral triangle and the Napoleon equilateral triangle. In more than ...
14
votes
3answers
871 views

An ellipse through 12 points related to Golden ratio

I am looking for a proof of the problem as follows: Let $ABC$ be a triangle, let points $D$, $E$ be chosen on $BC$, points $F$, $G$ be chosen on $CA$, points $H$, $I$ be chosen on $AB$, such that $IF$...
2
votes
0answers
146 views

Extension of a rational section of a projective bundle

Let us assume that we work over the complex field and let $X$ be a smooth projective variety and $\pi: P \to X$ a projective bundle (i.e. a fibration in projective spaces of constant dimension). Let $...
3
votes
2answers
200 views

Stability and complete types (in Model Theory)

I read the following statement in these slides of Saharon Shelah: "$K$ is stable iff for every $M \in K$ there are only "few" complete types over $M$." About the notation: here $K$ consists of all ...
2
votes
0answers
99 views

Intersection number of two projective curves using the resultant and tangent lines

For my thesis, I'm working on the intersection of projective plane curves over $\mathbb{C}$. We define the intersection number of projective plane curves (see for example Gibson - Elementary Geometry ...
0
votes
0answers
67 views

Twisted sheaves on tower of $\mathbb{P}^n$

Take the projective space $\mathbb{P}^n$ over a ring $W$. We call $\mathcal{O}(q)$ the usual twisted line bundle. Now take the map $f: \mathbb{P}^n\to\mathbb{P}^n$ defined by $$[x_0,\ldots, x_n]\...
8
votes
2answers
205 views

Linear sections of Segre varieties and rational normal scrolls

In a projective space $\mathbb{P}^{k+2}$ consider two complementary subspaces $\mathbb{P}^1,\mathbb{P}^k$, and let $C\subset\mathbb{P}^k$ be a degree $k$ rational normal curve. Fixed an isomorphism $\...
0
votes
0answers
95 views

Non-degenerate varieties in projective space

Let $X\subseteq \mathbb{P}^n(K)$, where $K$ is an algebraic closed field, be a projective variety. $X$ is called non-degenerate if $X$ is not contained in any hyperplane. For a given variety $X$, is ...
4
votes
1answer
104 views

Uniqueness of Mukai presentation of canonical model in genus 6

In his 92 paper, Mukai showed that a general genus $6$ curve may be represented in $\mathbb{P}^9$ as the intersection of the Grassmannian $G(2,5)$ (under the Plucker embedding), a plane $H\cong \...
5
votes
1answer
130 views

Rational map given by pfaffians

Consider a general skew-symmetric $(n+1)\times (n+1)$ matrix $Z$, and let su map $Z$ to the point of $\mathbb{P}^n$ determined by $[pf_0(Z):\dots:pf_n(Z)]$ where the $pf_i(Z)$ are the principal ...
2
votes
1answer
133 views

Very symmetric quadrangle in $\Bbb CP^2$

Is there a quadrangle $Q \subset \Bbb CP^2$, namely $Q$ is a set of four points, such that every permutation of $Q$ can be realizad by an isometric projectivity of $\Bbb CP^2$? Clearly the analogous ...
5
votes
1answer
181 views

Ideal of the Spinor variety $S^{10}\subset\mathbb{P}^{15}$

The ideal of the $10$-dimensional Spinor variety $S^{10}\subset\mathbb{P}^{15}$ is generated by $10$ quadrics. Does anyone know a reference where these 10 quadratic equations are written down ...
2
votes
1answer
202 views

Closure of quasi projective scheme

If $S$ is a scheme, $X$ is a smooth quasi-projective $S$-scheme, is the $S$-projective closure of $X$ a smooth $S$-scheme with $X$ an open subscheme?
3
votes
0answers
165 views

Picard group of quasi-projective varieties

Let $X$ be a smooth open sub-variety of a projective, not necessarily smooth, variety $X'$, defined over a finite field. Is $\text{Pic}(X)$ a finitely generated abelian group? I'm tempted to just ...
5
votes
1answer
176 views

Intermediate moduli spaces of stable maps

In the following paper: A. Mustata, M. A. Mustata, "Intermediate moduli spaces of stable maps", Invent. math. 167, 47–90 (2007) the authors introduced a variation on moduli spaces of stable maps ...
6
votes
2answers
233 views

Rational normal curves and tangent lines

Let $C,\Gamma\subset\mathbb{P}^n$ be degree $n$ rational normal curves in $\mathbb{P}^n$, such that for any $p\in C$ the tangent line $T_pC$ of $C$ at $p$ is tangent to $\Gamma$ as well. This means ...
2
votes
0answers
113 views

On the classification of spherical varieties

Let $G$ be a connected reductive algebraic group, for instance take $G = SL_n$. Does there is a classification of the $\mathbb{Q}$-factorial normal projective varieties with given dimension and Picard ...
2
votes
0answers
115 views

Flip of moduli space of stable maps

Let $\overline{M}_{0,2}(G(3,7),4)$ be the moduli space of $2$-pointed degree $4$ stable maps to the Grassmannian of $3$-planes in $\mathbb{P}^7$. Consider the divisor $\Delta$ whose general point is a ...
2
votes
1answer
185 views

Compactifications of spaces of morphisms

Let us denote by $Mor_3(\mathbb{P}^1,\mathbb{P}^3)$ the spaces of degree three morphisms $f:\mathbb{P}^1\rightarrow\mathbb{P}^3$, $$f(x_0,x_1)=[f_0(x_0,x_1):f_1(x_0,x_1):f_2(x_0,x_1):f_3(x_0,x_1)]$$ ...
2
votes
0answers
127 views

Reference for the Koszul--Malgrange Theorem

The Koszul--Malgrange theorem, roughly, identifies holomorphic vectors bundles over a complex manifold, as those finitely generated projective modules admitting a flat $(0,1)$-connection. The ...
4
votes
1answer
193 views

Ring of sections and normalization

Let $D$ be a base-point-free divisor on a normal projective variety $X$, and let $Y$ be the image of the morphism $f_{D}:X\rightarrow Y$ induced by $D$. Assume that $f_D$ is birational. Now, let $X(D)...
3
votes
1answer
116 views

Flipping and flipped loci

Let $f:X\dashrightarrow Y$ be the flip of a small contraction $\phi:X\rightarrow Z$, and let $\psi:Y\rightarrow Z$ be the small contraction such that $\psi\circ f = \phi$. Let $Exc(\phi), Exc(\psi)$ ...
2
votes
0answers
158 views

On a class of loci in Chow varieties

Let $k$ be a field, $i:X\hookrightarrow \mathbf{P}(\mathscr{E})$ be a fixed projective embedding of a smooth projective $k$-variety $X$, whose dimension is pure and equals $d\ge 0$. For $0\le p\le d$,...
6
votes
1answer
196 views

Where do the (Akizuki)-Nakano Identities First Appear

The answers to this M.O. question give a history of the Kaehler identities. The identities can be extended to the vector bundle-valued setting, and play a central role in the proof of the Kodaira ...
3
votes
1answer
152 views

Curves contracted by a rational map

Let $D$ be a big but not nef divisor on a normal $\mathbb{Q}$-factorial projective variety. Assume that the section ring $$R(D) = \bigoplus_{n\in\mathbb{N}}H^0(X,nD)$$ is finitely generated and ...
7
votes
2answers
315 views

Infinite projective plane with small edges

Let $\kappa$ be an infinite cardinal. We say $E\subseteq {\cal P}(\kappa)$ is an infinite projective plane on $\kappa$ if $e_1\neq e_2\in E$ implies $|e_1\cap e_2| = 1$, and whenever $n\neq m\in \...
10
votes
1answer
211 views

G-modules and ideals of secant varieties

Consider the action of $G = SL(n+1)$ on $\mathbb{P}^N$, and embed $\mathbb{P}^n$ in $\mathbb{P}^N$ via the degree two Veronese embedding. Let $V\subset\mathbb{P}^N$ be the corresponding Veronese ...
5
votes
0answers
370 views

Variational Hodge Conjecture vs Hodge Conjecture

Motivation. Let us state the following version of Grothendieck's variational Hodge Conjecture: Conjecture (VHC). Let $\mathcal{X}\to S$ be a proper smooth map of smooth algebraic varieties over $\...
15
votes
1answer
296 views

Birational automorphisms of varieties of Picard number one

Let $X$ be a smooth projective variety of Picard number one, and let $f:X\dashrightarrow X$ be a birational automorphism which is not an automorphism. Must $f$ necessarily contract a divisor?
10
votes
1answer
473 views

$K_0$-equivalence of varieties

Let $k$ be an algebraically closed field of characteristic zero. Let $A$ be the $\mathbf{Z}$-subalgebra of the Grothendieck ring of $k$-varieties $K_0(\text{Var}_k)$ generated by classes of semi-...
6
votes
1answer
152 views

Blowing-up an ideal generated by squares

Let $f_1,\dots,f_r$ be regular functions on a smooth projective variety $X$, and consider the ideals $I = (f_1^2,\dots,f_r^2)$ and $J = (f_1,\dots,f_r)$. Let $Y = Z(I)$ and $W = Z(J)$ be the ...
1
vote
1answer
171 views

If $A\subset B$, what to say about their $\operatorname{Proj}$?

Let $k$ be a field and $A$ and $B$ be two graded $k$-algebras satisfying $A\subset B$. The $\operatorname{Proj}$ construction is not functorial but is there nothing to say about $\operatorname{Proj}(A)...