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
148
questions
3
votes
1
answer
262
views
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 ...
- 31
13
votes
2
answers
452
views
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 ...
- 435
1
vote
0
answers
102
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,
$$Y \rightarrow \mathrm{Spec}(H^0(Y,\mathcal{O}_Y))$$
is proper or projective?
- 1,537
0
votes
0
answers
68
views
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.
...
4
votes
1
answer
276
views
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 ...
- 4,994
2
votes
0
answers
135
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 ...
- 18.7k
1
vote
0
answers
79
views
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 ...
- 1,303
2
votes
1
answer
83
views
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 ...
- 1,303
4
votes
0
answers
188
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 ...
- 818
2
votes
0
answers
956
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} ...
- 935
4
votes
0
answers
155
views
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\...
- 101
1
vote
0
answers
80
views
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 ...
- 11
3
votes
0
answers
164
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 $X$...
- 337
4
votes
1
answer
174
views
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 ...
- 215
0
votes
0
answers
194
views
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 ...
5
votes
1
answer
333
views
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 $...
- 145
0
votes
0
answers
189
views
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-...
- 51
2
votes
0
answers
96
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 ...
- 169
4
votes
3
answers
237
views
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 ...
- 351
2
votes
0
answers
119
views
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 ...
- 143
1
vote
0
answers
186
views
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 ...
- 685
0
votes
0
answers
178
views
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 $Y$...
- 428
2
votes
0
answers
86
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(...
- 351
5
votes
2
answers
401
views
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\...
- 1,185
0
votes
1
answer
144
views
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$ ...
- 968
3
votes
0
answers
172
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 ...
- 747
6
votes
2
answers
570
views
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 ...
- 351
4
votes
1
answer
229
views
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 ...
- 1,483
3
votes
0
answers
83
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)$ ...
- 486
2
votes
0
answers
70
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 ...
- 111
1
vote
1
answer
395
views
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,...
- 73
2
votes
1
answer
266
views
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 ...
- 1,091
3
votes
1
answer
231
views
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 ...
- 131
4
votes
0
answers
76
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}$...
- 7,415
5
votes
1
answer
350
views
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}^...
- 100
0
votes
0
answers
103
views
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$....
- 4,994
0
votes
0
answers
87
views
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, ...
- 3,373
1
vote
0
answers
189
views
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 ...
- 4,994
2
votes
2
answers
219
views
Does a smooth cubic in $P^3$ have projectively isomorphic sections?
We will be working over an algebraically closed field of characteristic 0. We say that a projective variety $X\subset \mathbb{P}^n$ has projectively isomorphic plane sections if there is an open set $...
- 747
1
vote
0
answers
120
views
Globally generated sheaf arising from orthogonal Grassmannian
We will use the Grothendieck's notation, according to the book of Hartshorne.
Let us consider a finite dimension $\mathbb C$-vector space $V$, with a non-degenerate symmetric bilinear form $q:V \times ...
- 371
1
vote
0
answers
53
views
Vector bundle associated to orthogonal flag
Let $V$ be a $(2n+1)$-dimensional complex vector space endowed with a non-degenerate symmetric bilinear form $q:V\times V\to \mathbb C$.
Fix the notation:
$$
OG(n-1,n,V):=\{W_{n-1}\subset W_n\subset ...
- 371
1
vote
0
answers
91
views
Global sections of a vector bundle over $OG(2,7)$
Let us work over $\mathbb C$, using the Grothendieck projectivization $\mathbb P():=Proj(Sym())$.
Consider a $7$-dimensional vector space $V$ endowed with a symmetric non-degenerate bilinear form $q:V ...
- 371
1
vote
0
answers
145
views
Restriction of a line bundle on $G/B$ to a fibre which is isomorphic to $\mathbb{P}^1$
Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$, Borel $B \supset T$ and Weyl group $W$. Set $X:=G/B$ and $C_w:=BwB/B \subset X$ for $w \in W$ the ...
- 451
3
votes
1
answer
282
views
Compact hyperkahler manifold as algebraic variety in weighted projective space?
Many examples of Calabi-Yau manifolds are constructed as algebraic varieties in weighted projective space, or more generally as complete intersection Calabi-Yau (CICY) manifolds. Are there such ...
- 427
5
votes
1
answer
297
views
Unsplitting sequence of vector bundles
Let $V$ be a $n$-dimensional complex vector space. Using Grothendieck's notation, we define the Grassmannian $G(k,V)$ as the space of $k$-quotients of $V$ or, equivalently, as
$$
G(k,V)=\{ \mathbb P W ...
- 371
0
votes
0
answers
209
views
Tautological bundle and its dual
Let $X=\mathbb G(2,V)$ be the Grassmannian of $2$-planes in $V=\mathbb C^n$. We denote by $\mathcal S$ the tautological bundle on $X$.
In a paper there is written that "since $\mathcal S$ is a ...
- 371
2
votes
1
answer
163
views
Moduli spaces of horizontal curves
Let $f:X\rightarrow Y$ be a morphism of projective varieties. We may assume that $X$ and $Y$ are smooth, and $f$ is flat of relative dimension one. Fix an ample divisor $A$ on $X$.
I would like to ask ...
- 397
1
vote
0
answers
95
views
Projective bundles on quadrics
Let us fix the setup over $\mathbb C$. Let $V$ be a $n$-dimensional vector space endowed with a non-degenerate symmetric bilinear form $q: V \times V \to \mathbb C$.
We have that
$$
Q^{n-2}=OG(1,V)=\{\...
- 371
1
vote
0
answers
114
views
Quadrics into Grassmannian as zero locus of a section
Let $V$ be a $\mathbb C$-vector space of dimension $n+2$ with a symmetric bilinear non-degenerate map $q: V \times V \to \mathbb C$. We define
$$
G(k+1,V):=\{\mathbb PW \subset \mathbb PV : \dim W=k+1\...
- 371
2
votes
1
answer
406
views
Tangent bundle for orthogonal and isotropic Grassmannians
We will work over $\mathbb C$. Let us consider a $n$-dimensional vector space $V$, then we define the $k$-th Grassmannian as
$$
\mathbb G(k,V):=\{W \subset V : \dim W=k\}.
$$
Then consider a non-...
- 371