Questions tagged [projective-geometry]
The projective-geometry tag has no usage guidance.
615
questions
4
votes
0
answers
111
views
Projective planes over algebraically closed fields
Suppose I am given a projective plane $P \cong \mathbb{P}^2(k)$ over a (commutative) field $k$.
With "projective plane," I mean the point-line geometry (and not, for instance, the scheme): $...
6
votes
2
answers
481
views
Global sections of multiples of a divisor
Let $D$ be an integral divisor on a smooth projective variety $X$. Consider the multiples $mD$ of $D$ for $m\geq 0$. Clearly, $h^0(X,mD) = 1$ for $m = 0$.
Is there any example where $h^0(X,mD) = 0$ ...
2
votes
1
answer
418
views
A question on effective divisors
Let $X$ be a projective variety with two morphisms $f:X\rightarrow Y$ and $g:X\rightarrow Z$ with irreducible fibers of positive dimension. Assume that $Pic(X) = f^{*}Pic(Y)\oplus g^{*}Pic(Z)$. Then ...
4
votes
1
answer
470
views
Normal bundle to Veronese varieties $v_d(\mathbb{P}^n)$ into $\mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^n}(d)))$
I was searching for a response on the internet but I was not able to find out an explicit answer.
It is known that if $\mathbb{P}^n \subset \mathbb{P}^N$ is embedded linearly then the normal bundle $...
2
votes
1
answer
372
views
Nef and pseudo-effective divisors over non algebraically closed fields
Let $X$ be a projective variety over a field $K$. As a consequence of Kleiman's criterion, when $K$ is algebraically closed, we have that if $D$ is a nef divisor on $X$ then $D$ is pseudo-effective.
...
0
votes
1
answer
253
views
Pseudoeffective divisors on surfaces
Consider a minimal smooth conic bundle $S$ of dimension two. Assume that there are two curves $C,F$ on $S$ such that $C^2 < 0$ and $F^2 = 0$. Let $D$ be a pseudoeffective divisor on $S$ such that $...
3
votes
1
answer
177
views
Projective manifold whose anticanonical section is composed of two components
Let $M$ be a connected projective complex manifold with a smooth anticanonical divisor $D$ ($D \sim -K_M$).
In an answer to a previous question,
It is told that $D$ may have at most two components.
An ...
2
votes
0
answers
135
views
Reference for bi-octonionic projective plane $\left(\mathbb{C}\otimes\mathbb{O}\right)P^{2}$
I'm not well versed in projective geometry since it is not really my field. I read in [1] about the existance of a projective plane $\left(\mathbb{C}\otimes\mathbb{O}\right)P^{2}$ defined on bi-...
1
vote
1
answer
50
views
Random linear map contracting distances on the projective line
Let $P^1$ be the real projective line, identified with lines of $\mathbb{R}^2$. Let $\mu$ be a probability measure on invertible linear map of $\mathbb{R}^2$ and let $(A_i)$ be i.i.d. random variables ...
6
votes
2
answers
584
views
Geometric meaning of coherent sheaves $\mathcal{F} \otimes \mathcal{O}_{\mathbb{P}^n}(d)$ over $\mathbb{P}^n$
Maybe it sounds like a silly question but I'm not able to figure out in my head the geometric meaning of "twisting" vector bundles (or more generaly coherent sheaf) over $\mathbb{P}^n$.
I ...
3
votes
1
answer
471
views
Trivial subbundle of universal bundle on the Grassmannian $\mathbb{G}(k,n)$
I was looking at the following paper by Tango:
https://projecteuclid.org/journals/journal-of-mathematics-of-kyoto-university/volume-14/issue-3/On-n-1-dimensional-projectlve-spaces-contained-in-the-...
10
votes
0
answers
191
views
Metrization of projective manifolds
A modern take on Hilbert's fourth problem could be as follows:
Given a manifold $M$ with a flat projective structure (i.e., a $(PGL(n+1),\mathbb{RP}^n)$-structure), find all metrics for which the ...
2
votes
1
answer
158
views
Restriction of small transformations
Let $\phi:X\dashrightarrow Y$ be an elementary small transformation (isomorphism in codimension $1$) between normal and $\mathbb{Q}$-factorial projective varieties.
Then there are small contractions $...
11
votes
1
answer
412
views
Planes in Lagrangian Grassmannians
Let $LG(h,2h)$ be the Lagrangian Grassmannian of subspaces of dimension
$h$ of a complex vector space of dimension $2h$.
For instace, $LG(1,2)=\mathbb{P}^1$, and $LG(2,4)\subset\mathbb{P}^4$ is
a ...
4
votes
2
answers
438
views
Subvarieties of Lagrangian Grassmannians
Let $LG(n,2n)$ be the Lagrangian Grassmannian parametrizing Lagrangian subspaces (so of dimension $n$) of $\mathbb{C}^{2n}$. Then $LG(n,2n)\subset G(n,2n)$, where $G(n,2n)$ is the Grassmannian of ...
2
votes
1
answer
261
views
How to express this hyperbolic extension of the cross-ratio in terms of hyperbolic distances and volumes?
Given $4$ distinct points on the Riemann sphere, thought of as the sphere at infinity of hyperbolic $3$-space $H^3$, one may define the cross-ratio in the usual way. Note that the cross-ratio is the ...
1
vote
0
answers
121
views
Combinatorics of projective planes over commutative rings
An axiomatic projective plane is a point-line incidence structure with the following axioms:
any two distinct points are collinear (via a unique line);
any two distinct lines meet in a unique point;
...
4
votes
0
answers
103
views
Singularities of hypersurfaces in projective bundles
I am doing some calculation on a toy example from the question here.
Let $\mathbb P(E) \rightarrow \mathbb P^1$ be the projectization of the vector bundle $E = \mathcal O \oplus \mathcal O \oplus \...
2
votes
1
answer
336
views
What is a generic pencil?
In Voisin book "Hodge theory and Complex Algebraic Geometry 2". There is the following corollary
Corollary 2.10. If $X\subset \mathbb{P}^N$ is a smooth projective complex variety, then a ...
2
votes
0
answers
289
views
Hypersurfaces in projective bundles over $\mathbb P^1$
I am working on a suggestion of a comment here.
Let $E \rightarrow \mathbb P^1$ be a non-trivial vector bundle of rank $r$ with $\deg E =0$ and $\mathbb P(E) \rightarrow \mathbb P^1$ be its ...
3
votes
1
answer
376
views
Examples of CY fibrations over $\mathbb P^1$
We work over $\mathbb C$ and let us call a smooth projective vartiety $M$ as Calabi-Yau (CY) manifold if it has trivial canonical class and
$h^i(M, \mathcal O_M ) = 0$ for $0 < i < \dim(M)$.
In ...
1
vote
0
answers
297
views
Rational and rationally chain connected surfaces
A projective variety $X$ over the complex numbers is rationally connected if two general points of $X$ can be joined by a rational curve in $X$, and rationally chain connected if two general points of ...
1
vote
0
answers
263
views
Canonical bundle formula for CY fibration over $\mathbb P^1$ without multiple fibers
Let us call a smooth projective vartiety $M$ as Calabi-Yau (CY) manifold if it has trivial canonical class and
$h^i(M, \mathcal O_M ) = 0$ for $0 < i < \dim(M)$.
In this definition, a CY 1-fold ...
6
votes
2
answers
375
views
CY fibration over $\mathbb P^1$ without any singular fibers
Let's call a smooth projective vartiety $M$ as Calabi-Yau (CY) manifold if it has trivial canonical class and
$h^i(M, \mathcal O_M ) = 0$ for $0 < i < \dim(M)$.
In this definition, a CY 1-fold ...
5
votes
1
answer
415
views
Golden ratio as a property of conic section (is it known?)
I am looking for a proof of a discovery as follows:
Let $ABC$ be arbitrary triangle and $(\Omega)$ be an arbitrary circumconic of $ABC$ let $A'B'C'$ is its tangential triangle of $ABC$ respect to $(\...
1
vote
1
answer
340
views
Thirteen-point conic and four-point line, are they new?
We know that Five points determine a conic and Two Points Determine a Line. Here I found a simple construct of a conic through $7$ points (in PS I note that how the conic through thirteen points) and ...
13
votes
2
answers
2k
views
Is it a new discovery on conic section?
I discovered a problem in plane geometry (there are some nice special cases) as follows:
Let $ABC$ be a triangle and $\Omega$ be arbitrary circumconic. Let two points $A_b, A_c \in BC$, $B_c, B_a \in ...
3
votes
1
answer
447
views
Projective invariants of the plane and cross ratio
I am looking for a reference for the following admittedly imprecise statement:
Any projective invariant of n points in the projective plane may be
expressed as a function of well-chosen cross-ratios.
...
10
votes
0
answers
225
views
Projective planes over non-division rings
Is there a "right" notion of a projective plane over a general (unital, non-division) ring?
Let me explain what type of object I am looking for. Let $R$ be an arbitrary (not necessarily ...
1
vote
1
answer
271
views
Tangent space to spaces of maps
Let $B = \{x_1,\dots,x_{d-2},y_1,\dots,y_k\}$ be a subscheme of $d-2+k$ distinct points of $\mathbb{P}^1$, and $g:B\rightarrow \mathbb{P}^2$ be a morphism mapping $x_1,\dots,x_{d-2}$ to a fixed point $...
1
vote
1
answer
349
views
A question on linear projection of a smooth projective variety
Let $X$ be a smooth, projective $\mathbb{C}$-variety of dimension $n$. Fix a closed point $x \in X$ and an embedding of $X$ in $\mathbb{P}^m$ for some integer $m$. For a given $d$, denote by $\sigma_d ...
5
votes
1
answer
417
views
Jumping conics in Grassmannians
Let $Gr(1,n)$ be the Grassmannian of lines in $\mathbb{P}^n$, and $f:\mathbb{P}^1\rightarrow Gr(1,n)$ a morphism of degree two. The pull-back $f^{*}S$ of the tautological bundle $S$ on $Gr(1,n)$ ...
0
votes
1
answer
170
views
Moving general fibers of a fibration
Let $X$ be an irreducible projective variety over $\mathbb{C}$ admitting a morphism $\pi:X\rightarrow \mathbb{P}^1$ with connected fibers. We may assume that the general fiber of $\pi$ is smooth.
My ...
2
votes
0
answers
70
views
Geometrical meaning of a question from Marden
Let $T \in SL(2,\mathbb{C})$ be a normalised Möbius transformation. Then, $$|T(z) - T(w)| =|z-w||T'(z)^{\frac{1}{2}}||T'(w)^{\frac{1}{2}}|$$.
The above is an exercise from Outer Circles by Marden (ex. ...
3
votes
1
answer
342
views
Irreducibility of the base and of the general fiber
Let $f:X\rightarrow Y$ be a morphism of scheme over $\mathbb{C}$. Assume that $Y$ and the the general fiber $F_y = f^{-1}(y)$ of $f$ are irreducible.
Does there exists an irreducible component $X'$ of ...
2
votes
0
answers
126
views
Is the projective symmetry group of a polytope more general than its linear symmetry group?
Give a (convex) polytope $P\subset\Bbb R^d$ (the convex hull of finitely many points). Consider its linear and projective symmetry groups:
\begin{align}
\DeclareMathOperator{\Aut}{Aut}
\...
3
votes
1
answer
244
views
Moduli spaces and conic bundles
The moduli space $A_2(1,8)^{\operatorname{lev}}$ of $(1,8)$-polarized abelian surfaces with canonical level
structure has a structure of conic bundle over $\mathbb{P}^2$ with a curve of degree $4$ as ...
2
votes
0
answers
112
views
Canonical class & ring of projective space $\mathbb{P}^n$ in differential geometry
David Mumford remarks in his book Algebraic Geometry I, Complex Projective Varieties on
page 109 that the fact that the canonical ring $\oplus_{k=0}^{\infty} \Omega_{k, \mathbb{P}^n}$
of projective ...
2
votes
1
answer
146
views
Degenerations of hyperelliptic coverings
Take six distinct points $p_1,\dots,p_6\in\mathbb{P}^1$ and consider the double covering $f:C\rightarrow \mathbb{P}^1$ ramified over $p_1,\dots,p_6\in\mathbb{P}^1$. Then $C$ is a smooth curve of genus ...
2
votes
1
answer
67
views
Smallest subset in $P^2 \mathbf F_q$ which cannot be disjointed from itself by a homography
Let $q$ be a power of a prime and $S \subseteq \mathrm P^2 \mathbf F^q$ such that
$$ \forall g \in \operatorname{PGL}(3,q), gS \cap S \neq \emptyset.$$
Can it be that $\vert S \vert < 1+q$ ?
(I ...
9
votes
1
answer
383
views
Set theoretic equation for Veronese varieties
Consider the embedding $f:\mathbb{P}^n\rightarrow\mathbb{P}^N$ induced by the complete linear system of degree $d$ hypersurfaces of $\mathbb{P}^n$. Its image $V_{n,\,d}$ is degree $d$ Veronese variety ...
3
votes
1
answer
278
views
3-secant lines of a projective curve
Consider a smooth projective curve $C\subset\mathbb{P}^n$. Let $G(1,n)$ the Grassmannian of lines of $\mathbb{P}^n$. The variety $S_2(C)\subset G(1,n)$ parametrizing lines that are secant to $C$ (i.e.,...
4
votes
1
answer
477
views
Higher order inflection points
Consider a smooth plane curve $X\subset\mathbb{P}^2$ of degree $d$. We will say that $x\in X$ is an inflection point of order $s$ if the tangent line $T_xX$, of $X$ at $x\in X$, intersects $X$ in $x\...
2
votes
1
answer
193
views
Configuration of points on a plane curve
Let $C\subset\mathbb{P}^2$ be a smooth plane curve of degree six. On $C$ there are $21$ points given as the intersection points of two lines choosen among a set of seven lines. More precisely there ...
1
vote
1
answer
237
views
Picard groups of determinantal varieties
Consider a general $4\times 4$ matrix:
$$
X:=\left(
\begin{array}{cccc}
X_0 & X_1 & X_2 & X_3 \\
X_4 & X_5 & X_6 & X_7 \\
X_8 & X_9 & X_{10} & X_{11} \\
X_{12} &...
2
votes
1
answer
259
views
Help about "Varieties with small Dual Varieties" by L.Ein
I'm studying the paper "Varieties with small Dual Varieties" by L.Ein and in the construction he gives about the $10-$dimensional spinor variety $S_4 \subset \mathbb{P}^{15}$ I'm finding ...
3
votes
1
answer
271
views
Linear spaces secant to Veronese varieties
The following question makes sense in a more general setting but for sake of simplicity let me stick to a particular case.
Consider the degree three Veronese embedding $V\subset\mathbb{P}^9$ of $\...
4
votes
1
answer
205
views
Software computing dimension and degree
Assume a projective scheme $X_{k_1,\dots,k_r}\subset\mathbb{P}^n$ is given as the set of common solutions of homogeneous polynomials $F_1(x_0,\dots,x_n),\dots,F_s(x_0,\dots,x_n)$, where the $F_i$ ...
2
votes
0
answers
178
views
Quadrics tangent to lines
I think that the following must be a basic question in enumerative geometry.
Take a line $L\subset\mathbb{P}^3$. The quadric surfaces in $\mathbb{P}^3$ that are tangent to $L$ are parametrized by a ...
1
vote
1
answer
87
views
Vertices of 2 self-polar triangles lie on conic
I have conic $\gamma$ and two self-polar triangles $ABC$, $XYZ$ with respect to my conic. Why can I construct a one conic through $ABCXYZ$?