# 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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### Intersection in toric variety

In a toric variety $T$ of dimension $11$ I have a subvariety $W$ of which I would like to compute the dimension.
On $T$ there is a nef but not ample divisor $D$ whose space of sections has dimension $...

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### Existence of hypersurfaces in projective space not containing any lines

Let $k$ be a field. Under what assumptions on $k$ is the following true?
There exists a hypersurface $H$ of $\mathbb P_k^n$ such that $H$ does not contain any line?
By a hypersurface, I mean an ...

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### Can one talk abstractly about the linear independence of points on the symmetric product of copies of a flag variety?

If one considers the (complete) flag variety $F(\mathbb{C}^2)$ of $\mathbb{C}^2$, then this is biholomorphic to $\mathbb{P}^1_\mathbb{C}$, and thus the symmetric product of $d$ copies of $F(\mathbb{C}^...

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### Irreducibility of Białynicki-Birula cells

Let $X\subset \mathbb{P}^n$ be a smooth complex projective variety, and consider a non-trivial action of $\mathbb{C}^*$ on $X$. For any connected fixed component $Y$ of the fixed point locus, we may ...

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### Galois ascent - When is a variety a Weil restriction?

Let $L|K$ be a finite Galois extension of degree $d$ and $X$ be a variety over $K$.
Is there a simple criterion, similar to Galois descent, allowing to determine whether $X$ is the Weil restriction (...

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### Is the associated G/B fibration to a G-torsor projective?

Let $X$ be a smooth projective variety over $\mathbb{C}$, $G$ a reductive group, and $P \to X$ a $G$-torsor. Let $B \subset G$ be a Borel subgroup. Is the associated $G/B$ fibre bundle $$ Y=G/B \...

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### What is the relationship between determinantal varieties?

Let $X$ be an $m\times n$ matrix.For a given positive integer $t ≤ \min(m,n)$, we denote the determinantal ideal $I_t = I_t(X)$ generated by the t-minors.What is the relationship between $I_1,I_2,I_3,\...

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### Proj construction and nilpotent homogenous elements in graded ring

Let $A= \bigoplus_{n \ge 0} A_n$ be a commutative Noetherian graded ring and $f \in A_d$ a nonzero homogeneous element of degree $d>0$. The natural ring map $q:A \to A/(f)$ induces a well defined ...

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### Projective subvarieties are closed?

I want to show that projective subvarieties of a quasi-projective variety are closed. One possible solution should be the following:
Let $W \subseteq \mathbb{P}^n$ be a quasi-projective variety and $V ...

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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 ...

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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 ...

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### Degree three, codimension one subvarieties lying on a quadratic hypersurface

Let $H$ be an irreducible hypersurface in $\mathbb P^n$ of large-ish degree, say 14. This question is about subvarieties $V$ of $H$ such that
$V$ has codimension 1 in $H$ (i.e. $V$ has dimension $n-2$...

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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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### Hodge symmetry without $\mathbb{C}$ [duplicate]

If $k$ is a field of characteristic zero and $X$ is a smooth irreducible projective variety over $k$, then $X$ satisfy Hodge symmetry, meaning that
$$\dim H^p(X, \Omega_{X/k}^q) = \dim H^q(X, \Omega_{...

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### Normal bundle of veronese as iteration extension of symmetric powers

In this post, an answer claims that the normal bundle of the Veronese decomposes into a filtration, such that the associated graded is $$\bigoplus_{i=2}^d S^i T,$$ where $T$ is the tangent bundle. But ...

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### A question on "Ample subvarieties of algebraic varieties"

Corollary 3.3 in Chapter IV of "Ample subvarieties of algebraic varieties" by R. Hartshorne asserts the following:
Let $X$ be a smooth projective variety and $Y\subset X$ a smooth subvariety ...

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### Degree of the preimage of a variety

Let $V\subset \mathbb{A}^m$ and $W\subset \mathbb{A}^n$ be affine varieties defined over an arbitrary field. Let $f:V\to \mathbb{A}^n$ be a morphism given by polynomials of degree $\leq D$.
Is it true ...

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### Composition of Gysin and restriction maps on $\ell$-adic cohomology

I already posted this question on mathstackexchange there, but I figured that it may have more replies here.
I follow the notations of Milne's lectures notes on etale cohomology, most specifically ...

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### Varieties with disjoint prime divisors

I've been following the works of Totaro, Pereira, and Bogomolov/Pirutka/Silberstein about algebraic varieties over complex numbers with families of disjoint divisors. The last one generalizes results ...

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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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### 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 ...

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### Projection from point on line on quintic del Pezzo surface

Let $X\subset \mathbb{P}^5$ be a quintic del Pezzo surface embedded anti-canonically and suppose $X$ is smooth. Suppose further we are given a line $L\subset X$. After a suitable change of variables ...

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### Finite subschemes of projective bundles

$\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\Proj{\mathbf{Proj}}$Let $S$ be a Noetherian scheme and $X$ finite $S$-scheme. The finite morphism $X \to S$ is projective in the sense of the ...

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### Isomorphism of varieties in $\mathbb{C}$ implies isomorphism over finite fields

Suppose that $X$ and $Y$ are algebraic varieties over $\mathbb{Z}$ (add your favourite hypotheses, like smooth or affine if needed). Denote by $X_k$ and $Y_k$ their base-change to varieties over a ...

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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?

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### Image and fibers of "nearly proper" morphisms

Let $X$ be a quasi-projective variety over $\mathbb{C}$, we say it is "nearly proper" if $X=Y-Z$ for some projective variety $Y$ and a closed subset $Z\subset Y$ of codimension at least two.
...

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### Universal hyperplane section and nondegeneracy of general hyperplane section

I have a question about Exercise 18.11 In Harris' book Algebraic Geometry, on page 231:
Give a proof of the nondegeneracy of the general hyperplane
section of an projective irreducible nondegenerated ...

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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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### Linear span of tangential variety

Let $X \subset \mathbb{P}^N$ be a projective variety of dimension $n$. Let us denote with $TX=\bigcup_{x \in X}\mathbb{T}_xX$ the tangential variety, where $\mathbb{T}_x X$ is the projective tangent ...

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### Linear system giving the projective embedding of the tangential variety

I was looking for a detailed explanation of a standard construction involving the projective tangential variety but I'm not able to find it anywhere, so maybe here some expert can enlight me on this ...

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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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### 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} ...

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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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### Lower bound of degree of ruled surface in $\mathbb P^n$

I have a question of Complex Algebraic surface in Beauville.
Let $S\subset\mathbb{P}^n$ be a (birationally) ruled surface of degree $d$ lying in no hyperplane.
Show that $d\geq 2 n-2$ if $S$ is not ...

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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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### Varieties connected by curves in projective spaces of small dimension

Let $X\subset\mathbb{P}^N$ be an irreducible complex variety. Fix an integer $a\geq 2$ and call $P_a$ the following property: given $x_1,\dots,x_a\in X$ general points there exists an irreducible ...

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### Irreducibility of plane algebraic curves

Given a plane algebraic curve
$$
y^n + a_1(x)y^{n-1} + \dots +a_{n-1}(x) + a_n(x)y = 0,
$$
with a branch point $P_0=(0, y_0)$ of order $n$. Can we prove that this curve is irreducible?
What if the ...

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### Does the main theorem of elimination theory with $\mathbb{Z}$-coefficients imply that projective varieties are complete?

I have a question concerning the completeness of projective varieties.
Let $k$ be an algebraically closed field. By the "main theorem of elimination theory" I mean the following result:
Let $...

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### Sheaves of abelian groups over a smooth projective variety

Can someone point some good reference (books or lecture notes) for these topics:
Let $X$ a smooth projective variety over an algebraically closed field
Sheaves of abelian groups over $X$
Quasi-...

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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 ...

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### Varieties with few trisecant lines

Let $X\subset\mathbb{P}^N$ be an irreducible projective variety. Let's denote by $\mathcal{T}$ the following property: through a general point $x\in X$ there is no line intersecting $X$ in at least ...

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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 ...

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### Meaning of "cut out (scheme-theoretically)"

Let $V$ be a projectively normal closed subvariety of some projective space over an algebraically closed field $\mathbb{K}$. Let $R$ be the local ring at the vertex of the affine cone over $V$ ($R$ is ...

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### Vector bundles on blowups

$\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\ker{ker}$Let $X$ be a (projective) variety, which is singular at a point $p$ of $X$ (we can also replace $p$ with some closed subvariety). Suppose $Y$...

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### Geometry of critical points of holomorphic maps in projective space

Let $f:\mathbb{CP}^n\to\mathbb{CP}^n$ be a holomorphic map; I am interested in what the subvariety of critical points could be.
More specifically, let $J=\{p\in \mathbb{CP}^n\ :\ \det\mathrm{Jac}(f)=0\...

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### A variation on the projective Nullstellensatz

Let $V$ be a $\mathbb{C}$-vector space, and let $f_1,\dots,f_n \in S^d(V^*)$ be homogeneous polynomials of degree $d$ for which $V(f_1,\dots, f_n)=\{0\}$.
Must there exist a positive integer $k\geq d$ ...

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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 ...

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### Smooth complete intersections

Let $X_{2,3}\subset\mathbb{P}^n$, with $n\geq 5$, be a complete intersection of a quadric $X_2$ and a cubic $X_3$ containing a $2$-plane $H$. Assume $X_2$ and $X_3$ to be general among the ...

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### Irrationality of some threefolds

Consider a smooth projective threefold $\overline W$, constructed in section 4 of this paper.
This threefold is a resolution of singularities of the quotient of a product of a K3 surface and $\mathbb ...

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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 $\text{Cl}(X)$ ...