Questions tagged [projective-varieties]
In algebraic geometry, a projective variety over an algebraically closed field $k$ is a subset of some projective $n$-space $\mathbb P^n$ over $k$ that is the zero-locus of some finite family of homogeneous polynomials of $n + 1$ variables with coefficients in $k$, that generate a prime ideal, the defining ideal of the variety
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questions with no upvoted or accepted answers
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Mathematical pendulum and $\mathbb C P^n$
I am very puzzled by the following remark on p.346 in Arnold's book "Mathematical methods of classical mechanics":
Another method of construction the same symplectic structure on complex ...
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Reference for the multiprojective Nullstellensatz?
Didn't get a single comment in over a day at math.SE, so maybe the question is more appropriate here.
I'm looking for a reference to a generalization of Hilbert's Nullstellensatz to the ...
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What can be said about the topological K-theory of non-singular varieties of small codimension in projective space?
Working over $\mathbb{C}$, the Barth-Larsen results tell us a lot about the ordinary cohomology of non-singular varieties of small codimension in projective space. For example if $X \subseteq \mathbb{...
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Find an explicit quasi-smooth embedding $X_{38} \subset \mathbb P(5, 6, 8, 19)$
This question is not quite about research-level mathematics, so I apologize for bringing it here. I asked it in Math.SE first, but I got no answers, and only a suggestion to ask it here.
Consider the ...
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Global sections of canonical line bundle on projective curve with everywhere vanishing derivative
Let $k$ be an algebraically closed field of positive characteristic $p$, $C$ be a curve (projective, non-singular, connected) of genus $g\geq 2$ over $k$ and $\omega \in H^0(C, \Omega_C)$ be a regular ...
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Description of determinantal varieties in $\mathbb{P}^n$ that are linear sections of determinantal varieties in $\mathbb{P}^{n+1}$
Fix an algebraically closed field $k$ of characteristic 0. Consider an $n$-tuple $(A_1,\ldots, A_n)$ of
$n\times n$ matrices over $k$ and assign to it the determinantal surface in $\mathbb{P}_k^{n-1}$ ...
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Berkovich Integration on algebraic curves
Berkovich developed a theory of integrating one-forms on his analytic spaces in his book "Integration of One-forms on $P$-adic analytic spaces". As this book is difficult to digest for me, I ...
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Equivariant projective embeddings with optimal dimension
Let $X$ be a complex projective manifold, and $f\in Aut(X)$ an automorphism, which is linearizable, that is, can be extended to an ambient projective space ${\mathbb P}^m$. I am interested to find ...
4
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203
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Cartan–Remmert reduction of an algebraic variety
Let $V$ be a normal connected algebraic (say, quasi-projective) variety over complex numbers. Assume that underlying complex analytic space $V^\text{an}$ is holomorphically convex, and thus admits the ...
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Explicit formula for embedding of real projective spaces into Euclidean spaces
I am interested in representing vectors in $\mathbb{R}^n$ in a sign-invariant and efficient manner. That is, I am looking for a function
$$f:\mathbb{R}^{n+1}\rightarrow\mathbb{R}^d$$
such that for $v\...
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Curves not invariant by non-trivial projective automorphisms
Let $g\ge 0$, $d\ge 1$ be integers. We consider the Moduli space $H_{g,d}$ parametrising smooth irreducible closed curves $C\subset \mathbb{P}^3$ of degree $d$ and genus $g$. Let us denote by $U_{g,d}$...
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Finite maps to normal varieties have fibers with bounded number of points
Let $f\colon X\rightarrow Y$ be a dominant, finite, and proper map of normal varieties of degree $d$ over an algebraically closed field $k$. Let $y\in Y$ be any closed point.
Question. Is it true that ...
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Complex projective algebraic variety, moduli space of flat connections, and instantons
In Looijenga's work below, if I understand correctly, it shows that
Statement 1: At an algebraic variety, the moduli space of SU($N$) flat
connections on a 2-torus $T^2$ is given by the space of ...
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Relations between an projective variety and galois cohomology
Let $f_1, \cdots, f_k$ be homogeneous polynomials over $\mathbb{Q}[x_0, \cdots, x_n]$. They define an projective variety $X$ over $\mathbb{P}^n(\mathbb{C})$, namely their set of zeros $$X = Z(f_1, \...
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Why is the "wrong" definition of intersection of varieties the "right" one for generalized Bézout?
For ease of notation, define the degree of a variety to be the sum of the degrees of its irreducible components. The generalized Bézout theorem (due to Fulton and Macpherson) states that, for $V_1$, $...
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How many elliptic curves over a finite field have a square discriminant?
$\newcommand{\char}{\operatorname{char}}$Given a finite field $F_q$ with $q\equiv 1 \bmod 3$ and $\char(F_q)>3$, I need to figure out how many isomorphism classes of elliptic curves $E/F_q$ have a ...
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What does a character of a scheme mean?
Here is a soft question I met in the book Introduction to Grothendieck Duality Theory by Altman and Kleiman.
In Chapter I the proposition 2.1 uses a term called "a character of $X$" where $X$...
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Explicit description of wonderful compactification for PGL_3
Let $k$ be an algebraically closed field of positive characteristics. Let $X$ be the wonderful compactification of $PGL_3$ (see for example section 6 of "Frobenius Splitting Methods in Geometry ...
3
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240
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Learning about determinantal varieties
In my research I recently stumbled upon a problem which involves trying to identify whether a given projective variety is determinantal or, even stronger, determinantal of a particular form. For ...
3
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Detecting non-principal Weil divisors on normal varieties using curves
Let $X$ be a normal projective variety over an algebraically closed field $k$. Given any morphism $f:Y\to X$, there is a pullback homomorphism $f^*:\text{Cl}(X)\to\text{Cl}(Y)$, where $\text{Cl}(X)$ ...
3
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Consequence of the failure of Nagata's conjecture
A modern version of the Nagata's conjecture says that
$$
L_{N,t}:=f_{N}^{*}(-K_{\mathbb{P}^{2}})-t\sum_{j=1}^{N}E_{j}
$$
is Ample for any $t<\frac{3}{\sqrt{N}}$, where $f_{N}:Y_{N}\to \mathbb{P}^{2}...
3
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Functorial lift of certain vector bundles to the ambient projective space
Given an very ample line bundle $L$ on a projective variety we embed it into a projective space such that pullback of $\mathcal{O}(1)$ is $L$. Then we can identify the two. Consider the full sub-...
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How to write down an explicit equation of given degree yielding a smooth hypersurface in a projective space?
Let F be a field of positive characteristic $p$ and let $d,n$ be two positive integers.
Can we explicitly write down an equation defining a smooth hypersurface $X_d⊂\mathbb P^n_F$ of degree d ?
This ...
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Does the Jacobian ring of a weighted projective hypersurface determine it up to isomorphism?
Let $V = H^0(\mathbb{P}^{n+1}, \mathcal{O}(1))$. Then the Mather-Yau Theorem states (Proposition 1.1 in Generic Torelli for projective hypersurfaces, Donagi)
Theorem. If $f,g \in S^dV$ have the same ...
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249
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Projective space over general schemes as quotient
Consider a general scheme $S$ and the projective scheme $\mathbb{P}^n_S$ over $S$.
Question: Is it possible to construct it as a quotient of an (open? affine?) subscheme $W_n\subset \mathbb{A}^{n+1}...
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Kan liftings and projective varieties
Regard the following two bicategories:
$\operatorname{dg-\mathcal{B}imod}$, with objects dg categories, and morphisms categories from $C$ to $D$ being the categories of $C$-$D$-bimodules. Composition ...
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When is the pushout of projective varieties along embeddings a projective variety?
From Karl Schwede's paper "Gluing schemes and a scheme without closed points'', I know that there exists a pushout of schemes for closed embeddings.
Now, if I start in the projective world I would ...
3
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Existence of regular hypersurface sections
Let $X$ be a irreducible regular projective variety over $Spec(O_K)$ for some number field $K$. Is it known that there exists at least one hypersurface over $Spec(O_K)$ such that cuts $X$ in a regular ...
3
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Coordinate ring of complete intersection Calabi Yau (CICY)
EDIT: If the question is for SE level just delete from here as it is also posted there. In fact I have seen some questions in SE regarding the coordinate rings of product of projective varieties but ...
3
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Given an embedding of $X$ into $\mathbb{P}^n_K$, do you get an induced embedding of any twist of it into $\mathbb{P}^n_K$?
Let $X$ be a projective algebraic curve over some number field $K$, and let $\varphi:X\hookrightarrow \mathbb{P}^n_K$ be an embedding of it (defined over $K$) into some projective space.
Now let $X'$ ...
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Grassmannian containing tangent variety of a curve
We work over $k=\mathbb{C}$. We consider the
the Grassmanian $G(2,4)$ of lines in $\mathbb P^3$ which we embed
by Plücker into $\mathbb P^5$. It is basic that under this embedding
$G(2,4)$ is ...
2
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190
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How dense is the set of rational points of a variety?
General question: Let $W$ be a proper subvariety of an irreducible affine variety $V/K$. Under what conditions do we know that $W(K)$ is a proper subset of $V(K)$?
If $K$ is finite, then one can bound ...
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When is a subspace of the cohomology of a smooth projective scheme on $k$ a motive?
Let $X$ be a smooth projective scheme over a number field $k$, and $V_{p}$ (resp. $V_{\text{dR}}, V_{\text{B}}$) a sub-space of $H_{et,p}^{\ast}(X)$ (resp. $H^{\ast}_{\text{dR}}(X), H^{\ast}_{\text{B} ...
2
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Control on the locus of bad reduction for divisors
Let $X$ be a smooth variety over a number field $K$ and let $\mathcal X$ be a normal, projective model of $X$ over the ring of integers $O_K$.
Now assume that $D\subset X$ is an irreducible divisor ...
2
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$0$-dimensional intersection in weighted projective space
Consider homogeneous polynomials $P_0,P_1,P_2,P_3,P_4,P_5$ of degrees $3,3,2,3,2,1$ over $\mathbb{P}^3$, and the map $\phi:\mathbb{P}^3\rightarrow\mathbb{P} = \mathbb{P}(3,3,2,3,2,1)$ given by
$$
\phi(...
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Is there any research about secant varieties by using homotopical algebra or simplicial methods?
Let $A$ be a finitely generated $\mathbb{C}$-algebra with $proj A$ is a projective variety $X \subset \mathbb{P}^n$. The join $J(X,X)$ can be represented by $proj (A\otimes_kA)$ and also we can ...
2
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Synthetic construction of rational normal curve
We consider the so called 'Synthetic or Steiner construction', which
can be found e.g. in this script or Joe Harris' Algebraic Geometry
on page 14 which should finally be recognized as rational normal ...
2
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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 ...
2
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High direct image of dualizing sheaf
I'm reading the paper "High direct image of dualizing sheaf" of professor Kollar. I summarizing my questions as follows:
Let $f:X\rightarrow Y$ be surjective projective morphism between ...
2
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200
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Clemens-Griffiths component birational invariant
Let $X$ be a smooth variety over complex numbers $\mathbb{C}$, say a threefold for sake of better intuition. Is there any geometrical intuition behind the fact that
the Clemens-Griffiths component of ...
2
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0
answers
427
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Embedding Calabi-Yau manifolds in projective space
When studying homological mirror symmetry, a lot of work is done not in the setting of complex manifolds, but of smooth (quasi-)projective varieties, see e.g. a paper from Orlov. However, the actual ...
2
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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 ...
2
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What is the precise relationship between projective duality and the Radon transform?
The Radon transform I am referring to is the one appearing in Brylinski's paper on projective duality, using the incidence correspondence over projective space and its dual projective space, $R p_{2*} ...
2
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362
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Intersection of two quadrics in $\mathbb{R} P^5$
Is there an ''algorithmic'' way to get that intersection of two quadrics
$$x_1 y_1 -x_2 y_2 - z_1^2+z_2^2=0$$
and
$$x_2 y_1 + x_1 y_2 -2z_1z_2=0$$ inside $\mathbb{R}P^5[x_1:x_2:y_1:y_2:z_1:z_2]$ is ...
2
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answers
158
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Graded Betti numbers $\beta_{n,j}$ for points in $\mathbb{P}^n$
Let $S = \mathbb{C}[z_0, \dots, z_n]$, and let $X$ be a set of points in $\mathbb{P}^n$. I'm looking for references concerning results for the graded Betti numbers $\beta_{n,j}(S/I(X))$, i.e., the ...
2
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216
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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-...
2
votes
0
answers
180
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The isometry groups of flag manifolds
For any sequence of integers $0<n_1<...<n_k$, there is a flag manifold of type $(n_1, ..., n_k)$, which is the collection of ordered sets of vector subspaces of $R^{(n_k)}$ $(V_1, ..., V_k)$ ...
2
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0
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215
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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 ...
1
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99
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Is $K_F\cdot C\leq K_X\cdot C$ for a fibre $F\subseteq X$ containing the curve $C$?
This is a question that I originally posted on Math Stack Exchange. After a couple of days I have not received any comments or answers, and after thinking about it more I realize that this question is ...
1
vote
0
answers
135
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Affinization map of cotangent bundle is proper/projective?
Given a cotangent bundle of a projective variety $Y=T^*X,$ do we know that its affinization map,
$$Y \rightarrow \mathrm{Spec}(H^0(Y,\mathscr{O}_Y))$$
is proper or projective?