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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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 ...
Nikita Kalinin's user avatar
7 votes
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431 views

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 ...
Igor Makhlin's user avatar
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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{...
Oliver Nash's user avatar
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6 votes
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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 ...
isekaijin's user avatar
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6 votes
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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 ...
Fabian Ruoff's user avatar
5 votes
0 answers
181 views

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 $\mathbb{P}_k^{n-1}$ ...
Sergey Guminov's user avatar
5 votes
0 answers
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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 ...
Fabian Ruoff's user avatar
4 votes
1 answer
176 views

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 ...
Misha Verbitsky's user avatar
4 votes
0 answers
203 views

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 ...
 V. Rogov's user avatar
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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\...
Felix Crazzolara's user avatar
4 votes
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78 views

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 $U_{g,d}$...
Jérémy Blanc's user avatar
4 votes
0 answers
298 views

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 ...
Yosemite Stan's user avatar
4 votes
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165 views

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 ...
wonderich's user avatar
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4 votes
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114 views

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, \...
HASouza's user avatar
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355 views

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 $V_1$, $...
H A Helfgott's user avatar
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3 votes
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119 views

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 ...
Jorge's user avatar
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3 votes
0 answers
190 views

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 $X$...
XYC's user avatar
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3 votes
0 answers
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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 ...
Asav's user avatar
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3 votes
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240 views

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 ...
Sergey Guminov's user avatar
3 votes
0 answers
102 views

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 $\text{Cl}(X)$ ...
Jonathan Love's user avatar
3 votes
0 answers
368 views

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}...
Trusio's user avatar
  • 71
3 votes
0 answers
149 views

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-...
user127776's user avatar
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3 votes
0 answers
196 views

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 ...
lefuneste's user avatar
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3 votes
0 answers
157 views

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 ...
alg_et_geom's user avatar
3 votes
0 answers
249 views

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}...
Nulhomologous's user avatar
3 votes
0 answers
163 views

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 ...
Markus Zetto's user avatar
3 votes
0 answers
227 views

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 ...
Arkadij's user avatar
  • 914
3 votes
0 answers
180 views

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 ...
user127776's user avatar
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3 votes
0 answers
296 views

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 ...
Martin Hurtado's user avatar
3 votes
0 answers
146 views

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 $X'$ ...
Quinlan Aktaş's user avatar
2 votes
0 answers
98 views

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 ...
JackYo's user avatar
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2 votes
0 answers
190 views

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 ...
H A Helfgott's user avatar
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2 votes
0 answers
1k views

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} ...
Marsault Chabat's user avatar
2 votes
0 answers
98 views

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 ...
manifold's user avatar
  • 299
2 votes
0 answers
93 views

$0$-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(...
Puzzled's user avatar
  • 8,842
2 votes
0 answers
77 views

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 ...
Doyoung Choi's user avatar
2 votes
0 answers
204 views

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 ...
user267839's user avatar
  • 5,948
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 ...
Basics's user avatar
  • 1,821
2 votes
0 answers
284 views

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 ...
xin fu's user avatar
  • 613
2 votes
0 answers
200 views

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 ...
user267839's user avatar
  • 5,948
2 votes
0 answers
427 views

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 ...
Markus Zetto's user avatar
2 votes
0 answers
52 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
2 votes
0 answers
131 views

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*} ...
A.S's user avatar
  • 121
2 votes
0 answers
362 views

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 ...
Filip's user avatar
  • 1,617
2 votes
0 answers
158 views

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 ...
Svinto's user avatar
  • 294
2 votes
0 answers
216 views

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-...
annie marie cœur's user avatar
2 votes
0 answers
180 views

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 $R^{(n_k)}$ $(V_1, ..., V_k)$ ...
wonderich's user avatar
  • 10.3k
2 votes
0 answers
215 views

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 ...
54321user's user avatar
  • 1,706
1 vote
0 answers
99 views

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 ...
Dave's user avatar
  • 111
1 vote
0 answers
135 views

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?
Filip's user avatar
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