Questions tagged [projective-geometry]
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615
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Number of conditions imposed by fat points to a linear system
Let $|D|$ be the linear system of degree $d$ hypersurfaces in $\mathbb{P}^n$ having multiplicity at least $m$ at $s$ general points.
Then $|kD|$ is the linear system of degree $kd$ hypersurfaces in $...
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Status of an open question in Artin's "Geometric Algebra"
In Artin's book "Geometric Algebra", Chapter II, the author states some axioms for geometry (section 1) and then begins to prove some results about the symmetries of the geometry (section 2).
The ...
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Monotone polygons (and polyhedra) with respect to a point
Dear mathoverflow community,
working on a visualization project I encountered a geometric problem, which I have not yet heard about and am interested in solving algorithmically. However a mere hint ...
2
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119
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Transversality of quadrics containing a projective curve
Let $C$ be a curve of genus $g$ and $L$ a $g^r_d$ on it and assume that we are in the range ${r+2\choose 2}>2d-g+1$. If $C$ and $L$ are chosen to be general then by the maximal rank conjecture (...
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When do two lines and three points determine exactly two conics? Exactly four?
In the real projective plane, I am given three points and two lines. I want to find out how many conic sections there are that are incident to each of the three points and tangent to each of the two ...
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Finding Equal-volume Triangulations in Homogenic Coordinates
Given the $n$-dimensional triangulation $\mathbb{T}^n$ of a finite set of points $\{p_1,\ ...\ ,\ p_{n+k}\} \subset \mathbb{R}^n$, is it always possible to find $n+k$ positive real weights $\{\omega_1,...
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Where can I find a proof of identity of $H^1(X,T_X)$ and a quotient by the jacobian?
I'm reading some notes on hodge theory by Charles Siegel which makes a claim on page 16 relating the space of deformations of a smooth projective hypersurface $X$ with the jacobian ideal. More ...
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Generalizing the fundamental theorem of projective geometry
A basic theorem of projective geometry states that any bijection that sends real projective $n$-space to itself and takes projective lines to projective lines is a projective transformation (it is ...
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704
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Curves and trisecant lines
We know that rational normal curves and elliptic normal curves have no trisecant lines. For the "next" case, this is still true. That is, a nondegenerate curve of degree $d\geq 5$ and genus $2$ in $\...
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What techniques are available for constructing D-modules over smooth projective varieties?
I'm trying to learn about D-modules for computing intersection cohomology but I'm having trouble coming up with explicit constructions of D-modules on projective varieties. Since this is an involved ...
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Intersection multiplicity of limit linear spaces
Let $X\subset\mathbb{P}^N$ be a smooth projective variety. Let us fix a general point $q \in X$, and let $C\subseteq X$ be a smooth curve passing through $q$.
Now let $\Lambda_{\xi, q}$, with $\xi \...
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A generalization of the Tucker circle theorem and the Thomsen theorem associated with a conic
I gave a generalization of the Tucker circle theorem and the Thomsen theorem at here. Now, I give a more generalization of these theorems as following:
Problem: Let $A_1A_2A_3A_4A_5A_6$ be a hexagon, ...
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Are there nonlinear projective spaces?
This is actually a series of questions posed by Guram Berishvili about the structure he calls marao. Everything I am going to write here I took (and messed up) from his home page which is all in ...
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Inverse Galois Problem ; Galois group of some branched cover of $P^{1}$ defined over $\mathbb{Q}$
I was trying to read a paper on Inverse Galois problem . I understands what the inverse Galois problem is. It asks if every finite group is the Galois group of some extension of the rationals.
The ...
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Approximating formal surfaces by analytic surfaces
Let $C$ be a smooth projective curve contained in a smooth projective $3$-fold (everything defined over $\mathbb C$). Let $\widehat X$ be the formal completion of $X$ along $C$ and suppose there ...
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Smooth curves in Tangent Developables
Let $C\subset\mathbb{P}^n$ be a smooth curve, and let $Y\subseteq\mathbb{P}^n$ be its tangent developable.
Given two general points $y_1,y_2\in Y$ does there exist a smooth curve $\Gamma\subset Y$ ...
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Cohomology of tangent sheaf of a singular hypersurface
Let $X\subset\mathbb{P}^n$ be a hypersurface singular at finitely many points $p_i\in X$. We may assume that $X$ has ordinary singularities at the $p_i$'s.
Does there exists a formula, perhaps in ...
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Elementary reference for the isometry group of $\mathbb{RP}^2$
Endow the real projective plane with the distance defined by $d(L,L')$ := "the angle between the lines $L$ and $L'$ ".
It is the case that every isometry from $RP^2$ onto $RP^2$ is induced by an ...
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Does there exist an Affinization or Projectivization process for Varieties?
Let us consider the classical isomorphism of real manifolds between $S^2$ and ${\mathbb CP}^1$. First strange thing we have here is that both are varieties, but $S^2$ is an affine and ${\mathbb CP}^1$ ...
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Sections of a linear system splitting as a product of degree one polynomials
Let $X\subset\mathbb{P}^n$ be a hypersurface of degree $d$ and with multiplicities $m_1,...,m_k$ at $p_1,...,p_k\in\mathbb{P}^n$ general points.
Let $S\subseteq |\mathcal{O}_{\mathbb{P}^n}(d)|$ be ...
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Are those $2$ quadratic forms congruent over $\mathbb{Z}[1/q]$
Let $q$ be a natural number (the first cases of interest being $q = 10,12$ or $15$), and let $n = q^2+q+1$. Also, let $I_n$ be the $n\times n$ identity matrix, and let $A_n$ be the $n\times n$ ...
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Quadrics passing through a point of a variety that are parametrized by a quadric
Let $X\subset\mathbb{P}^{N}$ be a $n$-dimensional algebraic variety and let $x\in X$. Let us suppose that
$$
\hat{Y}=\{\text{quadrics $Q\subset X$ of dimension $\frac{n}{2}$ such that $x\in Q$}\}
$$
...
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A chain of six circles associated with a conic
I found this problems three years ago. But I never have been a proof. Recently I posted in math.stackexchange.com. I am looking for a solution of the following problems:
A chain of six circles ...
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A new theorem in projective geometry
My question: I am looking for a proof of problem as following:
Introduction: When I research a theorem as following:
Theorem 1: Let $ABC$ be a triangle, let $(S)$ be a circumconic of $ABC$, let $P$...
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237
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Ideal of rational curve in projective space
Let $C_d\subset \mathbb P^d$ be a rational curve of degree $d>2$ ($C_d$ can be reducible) and $n\geq d$. Do we always have $h^0(I_{C_d}(n))=\binom {n+d}{d} - nd-1$?
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291
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Curves in homogeneous varieties
Let $C$ be a curve in a projective homogeneous variety $X$.
Fixed a general point $x$ in $X$, does there exist a curve $V$ in $X$ passing
through $x$ and such that $C$ and $V$ have the same homology ...
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765
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Hilbert metric for the Euclidean plane, a reference?
The Euclidean plane can be seen as an affine chart of the projective plane, together with a scalar product on the line at infinity, so that the absolute consists of two conjugate complex points at ...
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rationality of Fano 3fold $X_{18}$
I need a reference for an explicit proof of the rationality of the Fano 3-fold $X_{18}$. By explicit I mean by a sequence of explicit birational transformations.
Thank you!
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Automorphisms of Cartesian products
Let us consider the Cartesian product $X^r$, where $X$ is a smooth projective variety. There is a subgroup $Aut_{\Delta}(X^r)\subset Aut(X^r)$ of automorphisms of $X^r$ mapping a $k$-dimensional ...
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Automorphisms of locally trivial fibrations
Let $f:X\rightarrow Y$ be a locally trivial fibration with a variety $F$ as the fiber. Here $X, Y, F$ are smooth, projective varieties.
Does any automorphism of $F$ induce an automorphism of $X$?
In ...
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Sixteen points circle - A conjecture on Möbius plane
The conjecture refer the reader about the Bundle's theorem configuration. (This conjecture from a note)
Consider the Bundle theorem configuration :
Points $A_1, A_2, A_3, A_4$ lie on a circle,
...
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Automorphisms of cartesian products of curves
Let $C$ be a smooth projective curve. Is it true that
$$\textrm{Aut}(C\times C)\cong S_2 \ltimes (\textrm{Aut}(C)\times \textrm{Aut}(C))$$
and in case, what would be a reference for this? Thanks.
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871
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Inverse image of a divisor
Let $f:X\rightarrow Y$ be a morphism with connected fibers between projective varieties (not necessarily flat). Let $D\subset Y$ be an irreducible divisor.
Let us look at the cycle $f^{-1}(D)\subset X$...
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Resolution of indeterminacies for a map to Grassmannian of planes
Let $X\subset \mathbb P_k^N$ be a $n$-dimensional smooth projective variety ($n\geq 2$) and $\phi_l:X^l\dashrightarrow Gr(l,N+1)$ ($l\leq n+1$) be the natural rational map which associates to a ...
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Rigid effective divisors
Let $D\subset X$ be an effective smooth divisor in a smooth projective variety $X$. Assume that $h^0(X,D)=1$. In particular $D$ spans an extremal ray of the effective cone of $X$.
Now, let $f:X\...
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Generalization of the rigidity lemma in birational geometry
Let $X,Y,Z$ be projective varieties, and let $f:X\rightarrow Y$, $g:X\rightarrow Z$ be dominant morphisms. Assume that all the fibers of $g$ have the same dimension and are connected.
If there exists ...
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Degree of quasi-projective variety [closed]
Why we cannot define the degree of a quasi-projective $k$-variety ($k=\bar k$) $X$ for a given embedding $X\subset \mathbb P^n_k$ ?
If we take any compactification $\bar X$ of $X$, $\bar X\backslash X$...
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Determine the representation given by space of sections of symmetric products of cotangent bundle of projective plane
In a recent project, it was interesting for me to determine the $PGL(3)$ representation given by $H^0(S^2(\Omega(1)) \otimes \mathcal O(4))$ on $\mathbb P^2$. I did this by using the Euler sequence, ...
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Automorphism group of a variety
Suppose $X$ is a (quasi-projective) variety over a field $k$, and let $\mathbb{P}^n(k)$ be the ambient projective space.
When can one decide that the automorphism group of $X$ is induced by a ...
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Generalizing von Staudt's synthetic construction of the complex numbers
Starting from the real projective plane described synthetically/axiomatically, it is possible to construct the complex projective plane directly without passing through coordinates: one adjoins two ...
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Sections of the conormal bundle
Let $X\subset\mathbb{P}^N$ be a quadratic manifold. That is $I(X)$ is generated by quadratic polynomials $Q_1,...,Q_m$.
Let $\mathcal{I}_X$ be the ideal sheaf of $X$ and $\mathcal{I}_X/\mathcal{I}_X^...
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Automorphisms and infinitesimal deformations of a smooth complete intersection
Let $X\subset\mathbb{P}^{n+c}$ be a smooth complete intersection of dimension $n$.
Is it known when $Aut(X)$ is finite ?
Does there exist a formula for the dimension of the tangent space to the ...
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Sections of a sheaf of differentials on a weighted complete intersection
Let $X\subset\mathbb{P}(a_0,...,a_N)$ be a smooth $n$-dimensional weighted complete intersection in a weighted projective space $\mathbb{P}(a_0,...,a_N)$.
Is it true that if $q\geq 1$ then $H^0(X,\...
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Quartic symmetroids and 10-points sets
A quartic surface in $\mathbb{P}^3$ is said to be a "symmetroid" if its equation is obtained as the determinant of a 4x4 symmetric matrix of linear forms. It is well known that the general symmetroid ...
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Self intersection and deformations
Suppose I have a smooth projective surface $S$ and a curve $C\subset S$. The self-intersection of $C$ is by definition the degree of the restriction to $C$ of the normal bundle of $C$ inside $S$.
By ...
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A question on young persons guide to canonical singularities
In the Corollary at pag 407 of Young persons guide to canonical singularities there is a formula to compute the contributions $c_q(D)$ to Riemann-Roch of a divisor $D$ passing through a point $q\in X$,...
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Synthetic projective lines
The classical synthetic notion of projective plane consists of a set of points, a set of lines, and a relation of incidence between the two, such that any two distinct points lie on a unique line and ...
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Which weighted projective spaces (and their finite quotients) are local complete intersections?
Let $G$ be a finite subgroup of $\textrm{Gl}_{n+1}(k)$ (where $k$ is an algebraically closed field). My question is: do there exist examples of $G$ such that the corresponding quotient $P$ of $\mathbb{...
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Cross sections of semialgebric sets
Let $n$ be bigger than two, and let $A$ be a subset of the $n$-dimensional Euclidean space.
Suppose that the intersection of $A$ with any $(n-1)$-dimensional affine hyperplane is semialgebraic.
...
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135
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When is an intersection of quadrics empty
Is there an easy way to tell when the intersection of a set of quadrics (in complex projective space) is empty? Assume you know the quadrics explicitly.