The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. ...

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When are the sections of the structure sheaf just morphisms to affine space?

Let $X$ be a scheme over a field $K$ and $f\in\mathscr O_X(U)$ for some (say, affine) open $U\subseteq X$. For a $K$-rational point $P$, I can denote by $f(P)$ the image of $f$ under the map ...
4
votes
1answer
382 views

Koszul complex of locally free sheaves

Let $X$ be a complex variety; one can also assume it is smooth if this helps. $\mathcal{E}$ is a locally free sheaf of rank $r$ on $X$, and $s \in H^0(X, \mathcal{E})$. Then one has a Koszul complex ...
4
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2answers
115 views

How can we check the gluing property of sheaf of ideals?

For a ringed space $(X,\mathcal{O}_X)$, one can define a sheaf of ideals $\mathcal{J}$ of $\mathcal{O}_X$. Then how can we see the $\mathcal{J}$ satisfies the conditions of sheaf? Especially, I cannot ...
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1answer
95 views

Direct image for standart affine cover of projective line

Let $U_1 \cong Spec(K[t])$ and $U_2\cong Spec(K[t])$ be a standard affine cover of a projective line $\mathbb{P^1}(K)$, where $K$ is some field. Let us denote open embedding as $j_k$ $$ j_k : U_k \to ...
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1answer
140 views

Property kept under base change and composition is preserved by products

The following is true? Why? Let $P$ be a property of morphisms preserved under base change and composition. Let $X\to Y$ and $X'\to Y'$ be morphisms of $S$-schemes with property $P$. Then the unique ...
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1answer
125 views

Irreducibility preserved under étale maps?

I remember hearing about this statement once, but cannot remember where or when. If it is true i could make good use of it. Let $\pi: X \rightarrow Y$ be an étale map of (irreducible) algebraic ...
4
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1answer
749 views

Normalisation of an algebraic curve.

I need to compute explicitly the normalisation of a singular algebraic curve $C$ which is given by an explicit equation in $\mathbb{A}^2$. This task is mostly reduced to finding the integral closure ...
4
votes
1answer
651 views

Calculating the distance between a camera and a target using camera output

I have a 640x480 camera that recognizes a rectangle that is 1ftx2ft. Is it possible to calculate the distance between the camera and the rectangle? Edit: The horizontal angle of view is 54°.
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1answer
166 views

Basic Question on the Ideal-Variety Correspondence and Adjoint Functors

Let $k$ be an algebraically closed field. The ideal-variety correspondence says that the equations \begin{align} \mathbf{I}(X) &= \left\{f\in k[x_1,\dotsc,x_n]:p\in X\Rightarrow f(p)=0\right\} \\ ...
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1answer
192 views

Is a surjective homomorphism of regular local rings necessarily an isomorphism?

Let $R$ and $S$ be regular local rings, and $f: R\rightarrow S$ a surjection that induces an isomorphism on tangent spaces. Is $f$ necessarily an isomorphism? I believe the answer should be yes, ...
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2answers
424 views

Tangent spaces of affine algebraic varieties at singular points

Let $X$ be an affine algebraic over the algebraically closed field $k$ and let $\mathcal{O}(X)$ be the ring of its regular functions. Let us assume that $X$ is irreducible and let $x\in X$. There ...
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1answer
561 views

Why surjectivity stable under base change?

I want to prove that surjectivity is stable under base change: if $f:X\to S$ a surjective morphism of scheme and $\varphi:T\to S$ then $f_T:X\times_S T\to T$ is surjective. Idea 1: I know that for ...
4
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1answer
205 views

Is a morphism between schemes of finite type over a field closed if it induces a closed map between varieties?

This is the converse of this question. Let $X$(resp. $Y$) be a scheme of finite type over a field $k$. Let $f\colon X \rightarrow Y$ be a morphism. Let $X_0$(resp. $Y_0$) be the set of closed points ...
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1answer
120 views

Irreducible trivialization of a finite etale morphism

Let $X$ be an irreducible scheme and $Y \to X$ a finite étale morphism. Is there some finite étale cover $Z \to X$ which trivializes $Y$ (i.e. $Y \times_X Z$ is a union of copies of $Z$) such that $Z$ ...
4
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1answer
147 views

Cartier divisor and Dimension of Cohomology Group

I am doing some practice questions for my exam and I would appreciate help in solving this problem: $D,E$ are Cartier divisors on a nonsingular projective surface $X$. (1) If $D\equiv 0$ show ...
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1answer
55 views

Very ampleness of $\omega_{C}^n$

Let $C$ be a genus $g$ curve over complex numbers. How can I prove that $\omega_{C}^n$ is very ample for $n\ge2$ if $g=2$ and $n\ge 3$ if $g\ge 3$? Also, I wonder if this still true for other fields ...
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1answer
80 views

$\mathcal{O}_{X}(d)\simeq \mathcal{O}_{X}(D)$?

On $\mathbb{P}^n$ let $D$ be a smooth hypersurface defined by the equation $F=0$, F an homogeneous polynomial. $\mathcal{O}_{\mathbb{P}^n}(D)$ is the sheaf of meromorphic functions on $\mathbb{P}^n$ ...
4
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1answer
167 views

Is $Z(x^2-y^3)$ isomorphic to $Z(y^2-x^3-x^2)$ over the complex numbers?

I'm having trouble determining if the algebraic sets $Z(x^2-y^3)\subset \mathbb{A}^2$ and $Z(y^2-x^3-x^2)\subset\mathbb{A}^2$ are isomorphic over $\mathbb{C}$. My guess is that this boils down to ...
4
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1answer
428 views

When the the presheaf of image of morphism of sheaves is a sheaf?

For given a morphism of sheaves, in general, I know that the presheaf of image(or the presheaf of cokernel) is not a sheaf. is there when the the presheaf of image(or the presheaf of cokernel) of ...
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2answers
365 views

Example of the fixed point of a linear system?

Let $X$ be a normal projective variety and $D$ a Cartier divisor on it. A point $p\in X$ is called a fixed point of $|D|$ if $p \in \operatorname{supp}(D')$ for any $D'\in |D|$. Here $|D|$ is the ...
4
votes
2answers
324 views

Genus of curves embedded into some projective space

The Plucker formula allows one to calculate the genus of a curve embedded into $\mathbf{P}^2$. Does there exist a "higher-dimensional" analogue of the Plucker formula? More precisely, let ...
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1answer
115 views

If a principal divisor is defined over K, then is the function?

Let $X$ be an algebraic variety, $D$ a principal divisor of $X$ defined over $K$, i.e. the points of $D$ are in $X(K)$ and there is a function in $\overline{K}(X)$ whose divisor is $D$. Is $D$ ...
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1answer
281 views

Direct limit of localizations of a ring at elements not in a prime ideal

For a prime ideal $P$ of a commutative ring $A$, consider the direct limit of the family of localizations $A_f$ indexed by the set $A \setminus P$ with partial order $\le$ such that $f \le g$ iff ...
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1answer
93 views

When is a morphism of $S$-groupoids a monomorphism?

According to "Champs algébriques" by Laumon and Moret-Bailley, and $S$-groupoid is a category $\mathscr{X}$ and a functor $a: \mathscr{X} \to (\mathrm{Aff}/S)$, where $(\mathrm{Aff}/S)$ is the ...
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1answer
116 views

“standard” form of a finite morphism

Every etale morphism is locally (passing to affine neighbourhoods and then to their coordinate rings) of the form $A \to (A[x]/(P(x)))_b$ where $P(x)$ has the property that $P'(x)$ is invertible in ...
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1answer
308 views

Sheafification: another definition

Let be $\mathcal F$ a presheaf on a topological space $X$, the usual definition of the sheaf associated to $\mathcal F$ is the following $\mathcal F^+(U)=\{\widetilde s: U\rightarrow Et(\mathcal ...
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1answer
613 views

How is the hyperplane bundle cut out of $(\mathbb{C}^{n+1})^\ast \times \mathbb{P}^n$?

[Question has been updated with more context and perhaps a better explanation of my question.] Source: Smith et al., Invitation to Algebraic Geometry, Section 8.4 (pages 131 - 133). First, a brief ...
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1answer
157 views

Intersection numbers on a surface

Some (probably very easy) questions on intersection theory on surfaces... Say $S$ is a smooth projective surface over $\mathbb{C}$ with canonical divisor $K_S$. If $S$ is not ruled and $H$ is a ...
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3answers
372 views

Average degree of convex hull vertices in a Delaunay triangulation

Let $P \subset \mathbb{R}^2$. The boundary of $DT(P)$, the Delaunay triangulation of the point set $P$, is $conv(P)$. It is also known that the average degree of the vertices of $DT(P)$ is $\lt 6$. ...
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1answer
137 views

DVR-valued points of schemes

Let $X$ be a scheme of finite type over a discrete valuation ring $R$ with fraction field $K$, such that the generic fibre $X_K$ is smooth over $K$. Let $Y$ be a closed subscheme of $X$ which contains ...
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2answers
105 views

are fibres of a flat bijective map reduced?

Let $f: X \to Y$ be a flat map of algebraic varieties or of complex analytic spaces which is bijective on closed points (or just bijective in the secnond case). Suppose both $X$ and $Y$ are reduced. ...
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1answer
136 views

Number of birational classes of dimension d, geometric genus 0 varieties?

Fix an algebraically closed field $k$ and a positive integer $d$. My question is, what is the number of birational classes of dimension $d$, projective varieties over $k$ with geometric genus 0? If it ...
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1answer
97 views

Product of affine schemes

For any ring $A$, define a functor $\text{Spec}(A)$ from rings to sets by $$\text{Spec}(A)(R) = Hom_{\text{Rings}}(A,R)$$ Call a functor $X$ an affine scheme if it is isomorphic to a functor of the ...
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votes
4answers
352 views

Complement of a point in $\mathbb{P}^{2}$

This is question $5$ from Shafarevich's book page $66$. Let $X=\mathbb{P}^{2} \setminus x$ where $x$ is a point. Prove that $X$ is not isomorphic to affine nor a projective variety. How to prove this? ...
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1answer
641 views

Projective Nullstellensatz

I'm confused about the proof of the Nullstellensatz for projective varieties. If $J \subset k[x_0, \ldots , x_n]$ is a homogeneous ideal, we may regard $V(J)$ as a closed subset $ V(J) = V \subset ...
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1answer
126 views

Why is the domain of a rational function necessarily nonempty

Let $V$ be an irreducible affine variety. A rational map $f : V \to \mathbb A^n$ is an $n$-tuple of maps $(f_1, \ldots , f_n)$ where there $f_i$ are rational functions i.e. are in $k(V)$. Th map is ...
4
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1answer
329 views

Dimension of its irreducible components in Elimination Theory.

There is a small result I don't understand. To preface, for an algebraic variety $V\subset\mathbb{A}^n$ over some field $F$, one defines $\dim V=\operatorname{trdeg}(F(x)/F)$ for a generic point ...
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1answer
281 views

What does it mean to say a polynomial has an isolated singularity

In algebraic geometry, what does it mean when people say a polynomial $f$ has an isolated singularity at the origin?
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1answer
925 views

Quotient of an affine variety by a finite group

I have worked through the proof of the statement that a quotient of an affine variety X always exists in case the group G acting on X is finite (see "Algebraic Geometry, a First Course" by Harris, ...
4
votes
1answer
171 views

When is the canonical model of a curve nonsingular

Let $O$ be a Dedekind domain with fraction field $K$. Let $C$ be a smooth projective geometrically connected curve of genus $g>1$ over $K$. Let $p:X \to \mathrm{Spec} \ O $ be the canonical model ...
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1answer
196 views

$\pi^{tame}(\mathbb{A}^1_k)$ is trivial

Fixed an algebraically closed field of characteristic $p>0$, it is well known the result of the title: $\pi^{tame}(\mathbb{A}^1_k)\simeq 1$. Where the tame fundamental group, in this situation, ...
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1answer
227 views

Frobenius morphism and global sections of direct image of structure sheaf

Let $X$ be a proper scheme defined over an algebraically closed field of characteristic $p > 0$. Let $F : X\rightarrow X$ be the absolute Frobenius morphism. What is the dimension of $H^0(X, ...
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1answer
165 views

Different definitions of Kodaira dimension

Let X be a smooth projective variety with canonical class K. Let a be defined to be the maximum dimension of the image of X under the rational map induced by the linear system |nK| as n ranges over ...
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1answer
71 views

History and future of algebraic curves and the like?

Now that Fermat's last theorem has been proven, and also elliptic curves see widespread use in simple everyday applications, I would love to learn how the related theories came into beeing, how they ...
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1answer
54 views

Connected components functor for free coproduct cocompletions

Any extensive category admits a notion of connected object and hence a disconnected object. However, not all disconnected objects are presentable as disjoint unions of connected objects. Among ...
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1answer
110 views

Automorphisms of an elliptic curve fixing the invariant differential?

If we consider an elliptic curve $E/k$ given in Weierstrass form $y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}$, then I know that the translation maps $\tau_{P}$ with $P\in{E}$ fix the invariant ...
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1answer
39 views

Hartshorne 4.1.6 Gonality of a curve

I have a question about the following exercise from Hartshorne's book 'Algebraic geometry': Let $X$ be a curve of genus $g$. Show that there is a finite morphism $f:X\rightarrow \mathbb P^1$ with ...
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1answer
46 views

Hartshorne Lemma V.1.3 meaning of exact sequence

I've been trying to make sense of the exact sequence in Lemma 1.3 chapter 5. The Lemma is the following: Let $C$ be a smooth irreducible curve on a smooth projective surface X, and let $D$ be any ...
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1answer
75 views

Relation between ranks of free sheaves and cohomology

Suppose that $\mathbb{P}^r=\mathbb{P}^r_K$ is the projective space over a field $K$. Let $\mathcal{O}_{\mathbb{P}^r}(-1)^n\longrightarrow \mathcal{O}_{\mathbb{P}^r}^m$ be a morphism of vector ...
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1answer
37 views

Affine scheme obtained from (commutative) group algebra

Let $G$ be a finite abelian group (written multiplicatively), $R$ a commutative ring and let $R [G]$ denote the set of all formal linear combinations of elements of $G$ with coefficients in $R$. Then ...