Questions tagged [projective-geometry]

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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): $...
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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 $h^0(X,mD) = 0$ ...
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1 answer
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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 ...
Puzzled's user avatar
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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 $...
gigi's user avatar
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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. ...
Puzzled's user avatar
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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 $...
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3 votes
1 answer
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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 ...
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2 votes
0 answers
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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-...
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1 answer
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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 ...
Chevallier's user avatar
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 ...
gigi's user avatar
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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-...
gigi's user avatar
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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 ...
alvarezpaiva's user avatar
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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 $...
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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 ...
Elsa's user avatar
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2 answers
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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 ...
Puzzled's user avatar
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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 ...
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1 vote
0 answers
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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; ...
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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 \...
Basics's user avatar
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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 ...
Roxana's user avatar
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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 ...
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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 ...
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1 vote
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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 ...
user avatar
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 ...
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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 ...
Basics's user avatar
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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 $(\...
Đào Thanh Oai's user avatar
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 ...
Đào Thanh Oai's user avatar
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 ...
Đào Thanh Oai's user avatar
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. ...
coudy's user avatar
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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 ...
Anton Izosimov's user avatar
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 $...
Puzzled's user avatar
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1 vote
1 answer
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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 ...
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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 $Gr(1,n)$ ...
Puzzled's user avatar
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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 ...
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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. ...
Temari's user avatar
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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 ...
user avatar
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} \...
M. Winter's user avatar
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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 ...
Puzzled's user avatar
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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 ...
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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 ...
user avatar
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 ...
user102887's user avatar
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 ...
user avatar
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.,...
user avatar
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\...
user avatar
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 ...
user avatar
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} &...
user avatar
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 ...
gigi's user avatar
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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 $\...
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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 $F_i$ ...
user avatar
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 ...
user avatar
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 $ABCXYZ$?
Ivan Molotov's user avatar

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