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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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$$
such that for $v\...
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0
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Lower bound of degree of ruled surface in $\mathbb P^n$
I have a question of Complex Algebraic surface in Beauville.
Let $S\subset\mathbb{P}^n$ be a (birationally) ruled surface of degree $d$ lying in no hyperplane.
Show that $d\geq 2 n-2$ if $S$ is not ...
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What does a character of a scheme mean?
Here is a soft question I met in the book Introduction to Grothendieck Duality Theory by Altman and Kleiman.
In Chapter I the proposition 2.1 uses a term called "a character of $X$" where $X$...
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Varieties connected by curves in projective spaces of small dimension
Let $X\subset\mathbb{P}^N$ be an irreducible complex variety. Fix an integer $a\geq 2$ and call $P_a$ the following property: given $x_1,\dots,x_a\in X$ general points there exists an irreducible ...
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Irreducibility of plane algebraic curves
Given a plane algebraic curve
$$
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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$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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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 $\text{Cl}(X)$ ...
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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 ...
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Irreducible components of a projective variety
I would like to understand the irreducible components of a projective algebraic set.
Given an irreducible and homogeneous polynomial $H(w,x,y)\in \mathbb{C}[w,x,y]$ we define
$H_i(w,x_0,x_i):=H(w,x_0,...
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Are "transverse" hyperplane sections of nondegenerate irreducible projectice varieties always nondegenerate
Let $X \subseteq \mathbb{P}^n$ be a irreducible complex projective variety. It is called nondegenerate if it is not contained in a hyperplane in $\mathbb{P}^n$.
Assuming $X$ is nondegenerate and ...
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Explicit defining equations for del Pezzo surfaces
Are there known explicit parametrizations of smooth del Pezzo surfaces, say of degree $5$ or higher, as surfaces cut out by equations in some projective space?
The closest I've been able to find is on ...
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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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Square root of a line bundle up to a finite surjective morphism
Given a projective variety $X$ over a field of any characteristic, consider a line bundle $\mathcal{L}$ over $X$.
The existence of a line bundle $\mathcal{L}^\prime$ with an isomorphism ${\mathcal{L}^...
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Quotient $(V -S)/G$ is a quasi-projective variety for every closed $S \subset V$ with free $G$-action
I have a question about the statement of Remark 1.4 following on Thm 1.3 from Burt
Totaro's paper "The Chow Ring of a Classifying Space" (p. 4):
Let $G$ be a reductive group over a field $k$....
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The dimension of algebraic set $\{ F(x_1y_1, ..., x_my_m) = 0 \}$ compared to the dimension of $\{ F(x_1, ..., x_m) = 0 \}$
Let $F_i$ be homogeneous forms with $\mathbb{C}$ coefficients in $n$ variables for each $i$.
Let
$$
T_2 = \{ (\mathbf{x}, \mathbf{y}) \in \mathbb{A}_{\mathbb{C}}^{2n} : F_i(x_1y_1, ..., x_n y_n) =0, ...
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Rational normal curve as determinantal variety
We work here over complex numbers. Let $\Omega(Z)$
\begin{pmatrix}
L_1 & L_2 & ... & L_n \\
M_1 & M_2 & ... & M_n\\
\end{pmatrix}
be a $2 \times n$ matrix of homogeneous ...