Questions tagged [projective-geometry]

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Maximal abelian subgroups of the full collineation group $\mathrm{P\Gamma L}_3(q)$

Is there a convenient list of the maximal abelian subgroups of the projective semilinear group $\mathrm{P\Gamma L}_3(K) \cong \mathrm{PGL}_3(K) \rtimes \mathrm{Gal}(K)$ for $K$ a finite field? This is ...
Sean Eberhard's user avatar
2 votes
0 answers
183 views

Divisorial contraction to a non-normal variety

Consider a divisorial contraction $f:X\rightarrow Y$, between projective varieties, contracting an irreducible divisor $D\subset X$ to a subvariety $Z\subset Y$ of codimension at least two, and which ...
user avatar
1 vote
0 answers
126 views

Trisecant lines to curves in $\mathbb{P}^3$: a reference request

I'm starting to study the geometry of the locus of trisecant lines to a space curve $C \subset \mathbb{P}^3$. The main references, cited in almost all the papers, regarding the specific calculations ...
gigi's user avatar
  • 1,333
5 votes
2 answers
246 views

Surface of type $(2,2)$ on the Segre cubic scroll $\mathbb{P}^1 \times \mathbb{P}^2 \subset \mathbb{P}^5$

Let $S=\mathbb{P}^1 \times \mathbb{P}^2 \subset \mathbb{P}^5$ embedded with the Segre embedding given by $\mathcal{O}_S(1,1)$. If we intersect $S$ with a general smooth quadric $Q \subset \mathbb{P}^5$...
gigi's user avatar
  • 1,333
1 vote
0 answers
141 views

Cubic surface in $\mathbb{P}^3$ singular along a line

Maybe it is a stupid question but I'm not able to find the answer anywhere else. My goal is to prove in an "algebraic geometry fashion" that $\sqrt{n}$ is not a rational number for $n$ not a ...
gigi's user avatar
  • 1,333
0 votes
0 answers
92 views

Lines through the origin every pair of which meet at the same angle

This item isn't getting attention, so I'll try it here: begin quote The three lines through antipodal pairs of centers of faces of a cube meet each other pairwise at $90^\circ$ angles. The three lines ...
Michael Hardy's user avatar
1 vote
1 answer
222 views

Smoothness of moduli spaces of stable maps

If $X$ is a projective variety the moduli space of stable maps $\overline{M}_{0,0}(X,\beta)$ is a normal variety with finite quotient singularities. Can the pairs $(X,\beta)$ such that $\overline{M}_{...
user avatar
1 vote
0 answers
142 views

Kähler fiber space with base and fiber projective

Let $X$ be a Kähler manifold, $Y$ be a projective manifold, if $X$ exits a smooth fibration over $Y$ such that all the fibers are projective manifolds, then is $X$ a projective mannifold? If we do not ...
Tom's user avatar
  • 341
0 votes
1 answer
248 views

Does projective transformation preserve convexity? [closed]

Does projective transformation preserve convexity? Notice: Ignore the trivial case which projects a convex curve to a straight line.
Nan Zhang's user avatar
5 votes
1 answer
278 views

Isomorphisms of complete intersections

Let $X, Y\in \mathbb{P}^n$ be two singular Fano complete intersections of the same multidegree $(d_1,…,d_r)$. If we assume there is an isomorphism $f\colon X\rightarrow Y$ are there any assumptions so ...
Theodoros Papazachariou's user avatar
3 votes
0 answers
78 views

Infinite-dimensional quasifields

In their seminal paper on translation planes (The Construction of Translation Planes from Projective Spaces, Journal of Algebra 1:85-102, 1964, https://doi.org/10.1016/0021-8693(64)90010-9), Bruck and ...
Jeremy Dover's user avatar
3 votes
0 answers
163 views

Divisorial contractions and singularities

I have a smooth $6$-fold $X\subset\mathbb{P}^n$ and a divisor $D\subset X$ cut out by a quadratic polynomial. I know that $D$ in singular along a smooth $3$-fold $Y\subset X$, and that if $Z$ is the ...
user avatar
2 votes
1 answer
528 views

A question on a Macaulay2 computation

I have an ideal $I$ generated by quadratic and cubic homogeneous polynomials in $10$ variables. Macaulay2 tells me that $I$ defines an irreducible variety $X$ of dimension $5$ and degree $10$ in $\...
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11 votes
1 answer
1k views

What is the automorphism group of the projective line minus $n$ points?

$\DeclareMathOperator{\AGL}{\operatorname{AGL}}\DeclareMathOperator{\PGL}{\operatorname{PGL}}$What is the automorphism group of $\mathbb P^1$ minus $n$ points (let's say over an algebraically closed ...
Asvin's user avatar
  • 7,608
2 votes
0 answers
572 views

Embeddings of Hirzebruch surfaces $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$

Let $X_n=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$ be the $n-$th Hirzebruch surface. We know that for $d>0$ and higher $k>>0$ the linear system $$\mathcal{L}_{...
gigi's user avatar
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7 votes
1 answer
424 views

Strict transform of a tangent curve under blow-up

$\DeclareMathOperator{\Bl}{\operatorname{Bl}}$It is known that if we have a projective variety $X$ and a projective smooth subvariety $Y$ then the exceptional divisor $E \subset \Bl_{Y}X$ of the blow-...
gigi's user avatar
  • 1,333
4 votes
0 answers
213 views

Example of a computation of the volume of a subvariety in projective space $\mathbb{P}^n$

Let us consider the projective space $\mathbb{P}^n$ with the standard Fubini Study metric. I searched all over the internet but I can't find an example of a calculation of the volume for a projective ...
gigi's user avatar
  • 1,333
6 votes
0 answers
532 views

When a fibration over a projective manifold is projective?

Let $X$ be a complex manifold, $B$ be a complex projective manifold, consider a smooth fibration $\pi:X\rightarrow B$ such that all the fibers of $\pi$ are simply-connected projective manifolds, then ...
Tom's user avatar
  • 341
2 votes
0 answers
171 views

Understanding why $\pi_{3}(N) = \mathbb{Z}_{(2n, n^2)}$?

I am trying to understand the paper Arkowitz and Golasinski - Co-$H$ structures on Moore spaces of type $(G, 2)$: In section 4, titled "Homotopy elements of finite order", the authors say $\...
Smart20's user avatar
  • 121
3 votes
1 answer
192 views

Vanishing locus generic section $(\mathrm{sym}^2 \mathcal{R})(1)$

Let $n = 2m$ be an even integer and let $\mathcal{R}$ the tautological bundle on the Grassmannian $\mathrm{Gr}(2,n)$. I am looking for an explicit description of the degener The bundle $(\mathrm{Sym}^...
Libli's user avatar
  • 7,210
0 votes
0 answers
60 views

Does base point free linear system of polynomials generate higher degree polynomials

Let $k$ be an algebraically closed field. Let $S=k[x_0,\ldots, x_4]$ be the ring of polynomials. We set $S^i$ to the graded piece of degree $i$ polynomials. Let $H$ be a hyperplane of $S^5$ with no ...
user96145's user avatar
2 votes
0 answers
194 views

2 K3s and cubic fourfolds containing a plane

Two K3 surfaces show up when talking about cubic fourfolds containing a plane. Let $P\subset X\subset \mathbb{P}^5$ be the plane inside the cubic. Since $P$ is cut out by 3 linear equations then $X$ ...
IMeasy's user avatar
  • 3,707
13 votes
2 answers
558 views

A geometric definition of the addition law on abelian surfaces

Most people will have see a geometric "explanation" of the addition law on elliptic curves given by embedding it as a cubic in the projective plane and cutting it with lines. Is there a ...
Asvin's user avatar
  • 7,608
6 votes
1 answer
282 views

Singular cardinal $\kappa$ with projective plane such that all edges have cardinality $<\kappa$

Is there an infinite singular cardinal $\kappa$ such that there is a set $E\subseteq{\cal P}(\kappa)$ with the following properties? $|e| < \kappa$ for all $e\in E$, whenever $\alpha\neq\beta\in \...
Dominic van der Zypen's user avatar
0 votes
1 answer
45 views

Non-pencil infinite projective plane with edges of different cardinalities

A projective plane is a hypergraph $H=(V,E)$ such that if $e_1\neq e_2 \in E$ then $|e_1\cap e_2| = 1$, and for $v,w\in V$ there is $e\in E$ such that $\{v,w\}\subseteq e$. Is there a projective ...
Dominic van der Zypen's user avatar
4 votes
0 answers
114 views

Klein geometry associated to a degenerate conic

In his study of non-euclidean geometries, Felix Klein considers the group of projective transformations acting on the real projective plane whose extensions to the complex projective plane preserve a ...
coudy's user avatar
  • 18.5k
2 votes
1 answer
174 views

Decomposition of a morphism with positive dimensional fibers

It is well known that any birational morphism between projective varieties is a sequence of blow ups. Suppose now that I have a morphism $f:X \to Y$ with positive dimensional fibers, that is a ...
IMeasy's user avatar
  • 3,707
3 votes
0 answers
80 views

Presentations of $\mathbf{PGL}_3(\mathbb{F}_q)$ by three involutions

If I am not mistaken, the group $\mathbf{PGL}_3(\mathbb{F}_q)$, with $\mathbb{F}_q$ the finite field with $q$ elements, can be generated by three involutions. Where could I find such representations ? ...
THC's user avatar
  • 4,313
1 vote
1 answer
100 views

Presentations of $\mathbf{𝐏𝐆𝐋}_3(\mathbb{F}_2)$ by three involutions, 2

I am searching for a presentation of the group $\mathbf{PGL}_3(\mathbb{F}_2)$ for which the generators are involutions $a, b, c$, and such that the following relations are present [among extra ...
THC's user avatar
  • 4,313
1 vote
0 answers
170 views

Corollary of Mori’s theorem

Is there a direct proof of the corollary of Mori's theorem which says: If a projective complex manifold does not contain a rational curve then $ K_X $ is nef
Kamel's user avatar
  • 73
5 votes
2 answers
381 views

Presentations of $\mathbf{PGL}_3(\mathbb{F}_2)$ by three involutions

I am searching for (two) presentations of the group $\mathbf{PGL}_3(\mathbb{F}_2)$ for which the generators are involutions $a, b, c$, and such that the following relations are present [among extra ...
THC's user avatar
  • 4,313
1 vote
0 answers
231 views

Wonderful compactification of $\mathrm{SL}(2)/\mathrm{SO}(2)$

Let $\mathbb{P}^2 = \mathbb{P}(\operatorname{Sym}^2\mathbb{C}^2)$ be the projective space of $2\times 2$ symmetric matrices over $\mathbb{C}$ modulo scalar. Define an $\mathrm{SL}(2)$-action on $\...
user avatar
9 votes
1 answer
520 views

Degree of secant varieties of Veronese varieties

Consider the degree two Veronese embedding $\mathbb{P}^n\rightarrow\mathbb{P}^N$ and let $V^n_{2}\subset\mathbb{P}^N$ be the corresponding Veronese variety. Let $Sec_k(V^n_{2})\subseteq\mathbb{P}^N$ ...
user avatar
9 votes
0 answers
871 views

A new theorem (discovered in 2013) equivalent to Brianchon theorem (the old theorem) discovered in XIX century?

In 2013, I found a new problem as follows: Let six points $A_1$, $A_2$, ...$A_6$ lie on a circle $(O_1)$, and the six points $B_1$, $B_2$,...,$B_6$ lie on another circle $(O_2)$. If the quadruples $...
Đào Thanh Oai's user avatar
4 votes
1 answer
252 views

Quotient distance on $\mathbb{C}P^n$ is equivalent to distance induced by the Fubini-Study metric

Let $n$ be an even, positive integer. Then, is the metric (in the metric-space sense) on $\mathbb{C}P^n$ induced by the Fubini-Study (Riemannian) metric equivalent to the quotient (pseudo?)-metric ...
ABIM's user avatar
  • 4,989
1 vote
1 answer
207 views

Holonomy groups of Hermitian, and hyper-Hermitian, manifolds

An $n$-dimensional complex manifold $M$, endowed with an Hermitian metric $g$, is Kähler if and only if the holonomy group of $g$ is contained in $U(n)$. If $g$ is Hermitian but not Kähler do we ...
Fofi Konstantopoulou's user avatar
3 votes
0 answers
173 views

Hypothesis: An injection from polygons into $SO(2) \times S_n$

I have stumbled upon a possible representation of polygons by a concise description of their behaviour under rotation. I would like to know more about it and, obviously, if it is actually a bijection. ...
Robert Wegner's user avatar
1 vote
1 answer
113 views

How to find the optimal lines?

Does anyone know anyway or any algorithm that can exactly and/or numerically find lines $\left\{ l_{i}\right\} _{i=1}^{n_{k}+2}$ that maximizes $$\min_{1\le i<j\le n_{k}+2}\text{angle}\left(l_{i},...
Tanger's user avatar
  • 11
1 vote
0 answers
92 views

Infinite dimensional smooth projective geometry

Are there two infinite dimensional (Banach or Hilbert) manifolds $(P,L)$ which satisfy the axioms of a smooth projective geometry desribed in this page: Smooth Projectove Geometry
Ali Taghavi's user avatar
3 votes
0 answers
128 views

Isomorphisms of weighted complete intersections

Let $X\subset\mathbb{P}(a_0,\dots,a_n)$ and $Y\subset\mathbb{P}(b_0,\dots,b_n)$ be two weighted complete intersections with mild (say terminal) singularities. Assume that there is an isomorphism $f:...
user avatar
1 vote
1 answer
192 views

Terminal $\mathbb{Q}$-factorial divisorial contractions

Let $X$ be a $3$-fold, and $f:Y\rightarrow X$ a birational $\mathbb{Q}$-factorial divisorial terminal contraction (of relative Picard number one) contracting a divisor $E\subset Y$ to a point $p\in X$....
user avatar
2 votes
0 answers
51 views

Chow form of closure of product of affine varieties given the chow forms of their closurs

This question is about the connection between $\overline{X\times Y}$ and $\overline{X}$,$\overline{Y}$ where $X\subset \mathbb{A}^{n},\;Y\subset\mathbb{A}^{m}$ are affine varities over an ...
Espace' etale's user avatar
3 votes
1 answer
323 views

Moduli space of hyperplane sections of a projective variety

Let $k$ be a field and let $V$ a finite-dimensional vector space over $k$. Assume that $X\subset \mathbb{P}(V)$ is a closed subvariety. Does there exist a proper flat morphism $Y\to \mathbb{P}(V^*)$ ...
user avatar
3 votes
1 answer
416 views

Bertini theorem for connectedness

Let $X$ be a geometrically irreducible, possibly singular projective variety over an infinite field $k$. Assume that the dimension of $X$ is at least 2. Can there exist a hyperplane section of $X$ ...
user avatar
4 votes
1 answer
1k views

Ideal of rational normal curve of degree $d$

Let $A$ consist of the columns of the $2\times (d+1)$ matrix $$A=\begin{pmatrix} d & d-1 & \cdots & 1&0\\ 0 & 1 & \cdots & d-1 &d \end{pmatrix}$$ Then consider the ...
haziranyagmur's user avatar
6 votes
1 answer
283 views

Does any real projective plane incidence theorem follow from axioms?

Is it known whether any projective geometry statement that holds true in the real projective plane (equivalently, can be deduced from Hilbert axioms) follows from the standard projective axiomatics? ...
R. Matveev's user avatar
4 votes
0 answers
261 views

Understanding this example of projective geometry in Algebraic matroids

In this paper by Evans and Hrushovski: Projective planes in algebraically closed fields, they characterize projective planes in algebraically closed fields. These are coordinated by the skew-fields ...
A.B.'s user avatar
  • 407
2 votes
1 answer
290 views

Singular locus of a linear system of hyperplane sections

Let $X\subset\mathbb{P}^N$ be a rational smooth projective irreducible non degenerated variety of dimension $n=\dim(X)$ and let $$\mathcal{H}=|\mathcal{O}_X(1) \otimes\mathcal{I}_{{p_1}^2,\dots,{p_l}^...
gigi's user avatar
  • 1,333
2 votes
0 answers
160 views

A question on Okounkov bodies

Let $X$ be an irreducible $n$-dimensional projective variety, and $$Y_n\subset Y_{n-1}\subset\dots\subset Y_1\subset X$$ a flag of irreducible subvarieties such that $Y_i$ has codimension $i$ in $X$ ...
user avatar
4 votes
3 answers
476 views

Finding ellipse-ellipse intersections in $\mathbb R^2$

The setting is as follows: we considers 2 disks embedded in $\mathbb R^3,$ and are interested in projecting one disk (either) onto the plane of the other other, and then compute their area of ...
user929304's user avatar

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