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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59 views

dimension of linear system and multiplicity at a point

I recently encountered the following statement which I am unable to prove. Let $X$ be a smooth projective surface and let $L$ be a line bundle on $X$. For $x\in X$ if $h^0(|L|)\geq\frac{m(m+1)}{2}$ ...
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94 views

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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90 views

Nodes of rational plane sextic curves

According to the genus formula for plane curves, a nodal, irreducible rational plane curve $C\subset\mathbb{P}^2$ of degree six must have $10$ nodes. Now, if we take $10$ general points in $\mathbb{P}^...
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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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155 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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71 views

Determinantal representation of joins

Let $X^n = Z(I_X)\subset\mathbb{P}^N$ be an $n$-dimensional irreducible and non degenerate variety. Consider a linear subspace $H = Z(I_H)\subset\mathbb{P}^N$ of dimension $N-n-2$ disjoint from $X$. ...
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311 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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104 views

What is the étale fundamental group of projective spaces over finite fields?

Is there any convenient way to understand the étale fundamental group of projective spaces over finite fields, in particular, the étale fundamental group of $\mathbf{P}^2_{\mathbf{F}_q}$?
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132 views

Pencil of divisors in algebraic geometry

Let $X \subset \mathbb{P}^n$ be projective variety over alg closed field of char $0$ and $C = V(F), D= V(G) \subset X$ two distinct divisors (e.g. two quadrics, curves or lines lying in a surface,...) ...
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235 views

Degree of a variety vs degree of its blow-up

Let $V\subset \mathbf{P}^n$ be a variety. Let $f:\mathbb{P}^n\times \mathbb{P}^{n-1}\to \mathbb{P}^n$ be a blow-up of a point $P$ on $V$. Write $\widetilde{V}$ for the strict transform of $V$ under $f$...
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140 views

Generalization of universal sequence over Grassmannians

Setup: Let $V$ be a $(n+1)$-dimensional vector space. We define $\mathbb{P}V=\mathbb{P}^n$ as follows: points of $\mathbb{P}V$ correspond to 1-dimensional vector subspaces of $V^\vee$. Moreover, $$ ...
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145 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-...
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84 views

Injectivity of pushforward of rational Chow groups

I'd like to know whether there is a known counter-example to the following statement. Let $X$ be a smooth projective variety over a finite field. Let $Z$ be a codimension $2$ smooth subvariety which ...
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153 views

Linear spaces secant to Veronese varieties

The following question makes sense in a more general setting but for sake of simplicity let me stick to a particular case. Consider the degree three Veronese embedding $V\subset\mathbb{P}^9$ of $\...
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181 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 ...
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215 views

Chow's Lemma: Mumford's and Grothendieck's (?) definitions

David Mumford gives in his book Algebraic Geometry I, Complex Projective Varieties on page 61 a definition of Chow's Lemma which has at least for me not a usual form: If says that a closed $^*$-...
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466 views

Irreducible components: associativity for intersections?

Let $A$, $B$, $C$ be closed irreducible subvarieties of $\mathbb{A}^n$. Let $V_1$ be an irreducible component of $B\cap C$, and $V$ an irreducible component of $A\cap V_1$. Must there necessarily be ...
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109 views

Sections of vector bundles interpreted as sections of line bundles

Let $X$ be a smooth projective curve of genus $g$ over $\mathbb{C}$, $K_{X}$ be a cononical sheaf on $X$ and $\mathcal{E}$ be a locally free sheaf on $X$ s.t. $H^{0}(X,\mathcal{E}^{*})=\operatorname{...
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141 views

existence of birational morphism and divisors

The following result was metioned in a lecture: A nonsingular (or smooth) projective surface (variety of dimension 2) has a birational morphism to the projective plane, if and only if there exists an ...
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232 views

How many holes may a projection of an algebraic variety have?

Let $V$ be a closed subvariety of $\mathbf{P}^n$. (We work over an algebraically closed field.) Define $\pi:(\mathbf{P}^n\setminus P_0)\to \mathbf{P}^{n-1}$ by $\pi(x_0:x_1:...:x_n) = (x_0:x_1,...:x_{...
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112 views

Comparison of classical and Zariski topologies with constructible sets

In David Mumford's book Algebraic Geometry I, Complex Projective Varieties the proof of (3.25) Specialization principle on page 53 contains an argument I not understand. General assumptions: all our ...
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116 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 ...
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163 views

Unibranch points (definition for varieties over arbitrary field)

In David Mumford's book Algebraic Geometry I, Complex Projective Varieties treating mainly complex varieties as objects of interest on page 43 he defines what is a topologically unibranch variety $X$ ...
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179 views

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 ...
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5answers
487 views

Blow-up of projective variety $P^1 \times P^1… \times P^1$ ($n$ times) and blow-up of $P^n$

It is known that the blow-up of $P^1 \times P^1$ at a point is isomorphic to the blow-up of $P^2$ at two points. I'm wondering if there is any general statement for the blow-up of $P^1 \times P^1 \...
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1answer
209 views

Strict transform of a tangent curve under blow-up

$\DeclareMathOperator{\Bl}{\operatorname{Bl}}$It is known that if we have a projective variety $X$ and a projective smooth subvariety $Y$ then the exceptional divisor $E \subset \Bl_{Y}X$ of the blow-...
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97 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}...
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112 views

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 ...
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221 views

Question about Correspondences from Mumford’s Complex Projective Varieties

I study David Mumford's Algebraic Geometry I - Complex Projective Varieties and have some problems to understand a step in the proof of Lemma 6.7 (b). Firstly, the general setting & preparations ...
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196 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 ...
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249 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 ...
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150 views

Corollary of Mori’s theorem

Is there a direct proof of the corollary of Mori's theorem which says: If a projective complex manifold does not contain a rational curve then $ K_X $ is nef
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139 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 ...
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570 views

Recover the characteristic of $k$ from the category of $k$-varieties

Can you recover the characteristic of a perfect field from the category of smooth projective geometrically connected varieties over it?
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149 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 ...
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311 views

Degree of secant varieties of Veronese varieties

Consider the degree two Veronese embedding $\mathbb{P}^n\rightarrow\mathbb{P}^N$ and let $V^n_{2}\subset\mathbb{P}^N$ be the corresponding Veronese variety. Let $Sec_k(V^n_{2})\subseteq\mathbb{P}^N$ ...
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137 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 ...
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1answer
87 views

On relating $l(A), l(B)$ and $l(A+B)$ for Weil divisors on a smooth projective curve where one of the divisors is effective

Let $X$ be a smooth projective curve over an Algebraically closed field $k$. Let $k(X)$ denote its function field. If $A, B$ are Weil divisors on $X$ such that $A$ is effective (i.e. $A\ge 0$) , then ...
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139 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 ...
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108 views

About the multiplicity of intersection in projective space

I found this statement, and I can't get a proof: Given $X\subset \mathbb{P}^{n}$ an irreducible projective set of dimension $d$ and degree $e$, and given $H_{1},...,H_{d}$ hypersurfaces with degrees $...
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1answer
204 views

Non-unique completion of a flat family of smooth projective varieties

Let $\mathbb{k}$ be an algebraically closed field of characteristic 0. Denote $S=\mathrm{Spec}\:\mathbb{k}[t]$, $U=\mathrm{Spec}\:\mathbb{k}[t, t^{-1}]$, $Z=\mathrm{Spec}\:\mathbb{k}[t]/(t)$. What is ...
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182 views

Projective embeddings of quotients of normal varieties

Let $X$ be a normal complex projective variety of dimension $m$, $G$ be a finite subgroup of $\mathrm{Aut}(X)$, and $Y = X / G$ be the quotient. I am particularly interested in the case where $X$ is a ...
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1answer
291 views

Closed points of a closed subscheme of $\mathbb{P}^n$ over the residue field and the fraction field of a valuation ring $R$

Let $(R, M)$ be a valuation ring with algebraically closed fraction field $k$. Let $L = R/M$ be the residue field of $R$; it follows that $L$ is algebraically closed. I would like to understand the ...
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45 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 ...
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188 views

Bertini theorem for connectedness

Let $X$ be a geometrically irreducible, possibly singular projective variety over an infinite field $k$. Assume that the dimension of $X$ is at least 2. Can there exist a hyperplane section of $X$ ...
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171 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 ...
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131 views

Ample line bundle gives alternative description of a variety

Let $X$ be a (smooth) projective variety (over $\mathbb{C}$), and $\mathcal{L}$ an ample line bundle on $X$. I have heard that then $$ X \cong \mathrm{Proj} \left( \bigoplus_{k \ge 0} H^0(X,\mathcal{...
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87 views

Why can we require that the hyperdeterminant has integral coefficients?

Let $X=P^{k_1}\times \ldots \times P^{k_r}$ be the product of several complex projective spaces ($P^k$ is the projectivization of $\mathbb{C}^{k+1}$) in the Segre embedding into $P=P^{(k_1+1)\cdots(...
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2answers
267 views

When is a monomial rational map on the projective space birational?

Let $k$ be an algebraically closed field of characteristic $0$. For $\alpha :=(a_1,\dots,a_{n+1})\in \mathbb N^{n+1}_{\ge 0}$ , let $\bar x^{\alpha}:= x_1^{a_1} \dots x_{n+1}^{a_{n+1}} \in k[x_1,\...
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180 views

Is projective closure of a regular affine algebraic set also regular?

Now to specifics: Let $V \subset \mathbb{A}^3$ be a reducible affine algebraic set defined by two irreducible polynomials $f,g \in K[X,Y, Z]$ of degree $d$ ($K$ algebraically closed). So, if $V$ is ...