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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Projectively normal and normal when the ring satisfies Serre's criterion

In Hartshorne, Ex II 5.14, says that A closed subscheme $X \subseteq \mathbb{P}_{A}^r$ is projectively normal for the given embedding, if its homogeneous coordinate ring $S(X)$ is an integrally ...
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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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473 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 ...
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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 ...
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2 votes
2 answers
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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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257 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 ...
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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 ...
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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 ...
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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)=\{\...
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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\...
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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-...
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5 votes
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Question on a constructive proof that space projective curves are the intersections of three hypersurfaces

$\newcommand\P{\mathbb P} \newcommand\C{\mathcal C}$I am a bit confused by a proof I am reading on the fact that a projective algebraic space-curve (i.e. an algebraic curve in $\P^3(k)$, where $k$ is ...
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Secant variety to a zero-dimensional projective variety

This is a reference request/nomenclature question. Let $A \subseteq \mathbb{P}^n$ be a finite set of points not contained in a hyperplane (over some field), and let $\sigma_r(A)$ be the $r$-th secant ...
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2 votes
1 answer
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Do representations of same dimension implies isomorphic closed orbits?

Let us recall this fact. Let $G$ be a semisimple algebraic group over $\mathbb C$ and let $V,V'$ be two irreducible $G$-representations. We denote by $X,X'$ the unique closed $G$-orbits contained in $\...
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When are there no maps from a variety of high dimension to a variety of low dimension? [duplicate]

It's easy to show that the only maps from $\mathbb P^{n+d} \to \mathbb P^n$ are the constant maps for $d \geq 1$. Given two smooth, projective varieties $X,Y$ of dimensions $n+d,n$ as above, are there ...
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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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6 votes
3 answers
837 views

Open complement of hypersurfaces

Let $k$ be an algebraically closed field. Let $H_1, H_2$ be two smooth hypersurfaces of the same degree $d$ in $P^n_k$. Let $U_1,U_2$ be their complements respectively. Are $U_1,U_2$ isomorphic as ...
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Definition of rational equivalence

In the book "C. Voisin. Hodge Theory and Complex Algebraic Geometry. Volume II. Cambridge studies in advanced mathematics 76 (2002)" page 247, definition 9.4, says: Definition 9.4. The ...
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Grothendieck rings and the Tannakian formalism

I understand that the Tannakian formalism (which I only "know" extremely superficially) is very important for the theory of motives. I guess "the" conjectural category of motives ...
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Varieties swept out by Linear Spaces nondegenerated

We working over complex numbers $\mathbb{C}$ keeping our constructions as geometric as possible. Let $\Lambda_1, ..., \Lambda_m \cong \mathbb{P}^{n-2} \subset \mathbb{P}^{n} $ be pairwise distinct, ...
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2 votes
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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 ...
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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 ...
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1 vote
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Combinatorics of projective planes over commutative rings

An axiomatic projective plane is a point-line incidence structure with the following axioms: any two distinct points are collinear (via a unique line); any two distinct lines meet in a unique point; ...
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3 votes
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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 ...
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4 votes
1 answer
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Examples of CY fibrations over $\mathbb P^1$

We work over $\mathbb C$ and let us call a smooth projective vartiety $M$ as Calabi-Yau (CY) manifold if it has trivial canonical class and $h^i(M, \mathcal O_M ) = 0$ for $0 < i < \dim(M)$. In ...
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2 votes
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Canonical bundle formula for CY fibration over $\mathbb P^1$ without multiple fibers

Let us call a smooth projective vartiety $M$ as Calabi-Yau (CY) manifold if it has trivial canonical class and $h^i(M, \mathcal O_M ) = 0$ for $0 < i < \dim(M)$. In this definition, a CY 1-fold ...
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7 votes
2 answers
266 views

CY fibration over $\mathbb P^1$ without any singular fibers

Let's call a smooth projective vartiety $M$ as Calabi-Yau (CY) manifold if it has trivial canonical class and $h^i(M, \mathcal O_M ) = 0$ for $0 < i < \dim(M)$. In this definition, a CY 1-fold ...
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1 vote
1 answer
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showing a plane curve non-generic by exhibiting an even divisor

Let $G$ be a projective plane degree $2d$ curve with equation $AC-B^2$, with $A$, $B$, $C$ of degrees $\deg A=2d-4$, $\deg B=d$, $\deg C=4$. Then, for $d>2$, a dimension count shows that $G$ is not ...
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5 votes
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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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3 votes
1 answer
228 views

Irreducibility of the base and of the general fiber

Let $f:X\rightarrow Y$ be a morphism of scheme over $\mathbb{C}$. Assume that $Y$ and the the general fiber $F_y = f^{-1}(y)$ of $f$ are irreducible. Does there exists an irreducible component $X'$ of ...
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3 votes
0 answers
339 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}...
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