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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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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96 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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4answers
346 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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618 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 ...
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325 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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278 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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911 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, ...
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167 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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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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225 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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163 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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43 views

Projective curve $x^3+y^3=2z^3$ in $\mathbb P^2$ singular?

Is the projective curve $x^3+y^3=2z^3$ in $\mathbb P^2$ (defined over $\mathbb{C}$) singular or nonsingular? If singular, what are the types of these singularities? For an affine curve, one would ...
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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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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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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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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 ...
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79 views

Use Gröbner bases to count the $3$-edge colorings of planar cubic graphs…

I found a nice introduction on how to Use Gröbner bases to construct the colorings of a finite graph. Now my graphs $G=(V,E)$ are the line graphs planar of cubic graphs, so they are $4$-regular. The ...
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52 views

Is there an easy criterion to determine whether given polynomials form a complete intersection?

Suppose we have homogeneous polynomials in $s$ variables $F_1, ..., F_n$ with coefficients in integers. Let $X$ be a variety (or algebraic set) defined by the simultaneous equations $$ ...
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1answer
71 views

Extended ideals and algebraic sets

Let $L\subset k$ a field extension such that $k$ is algebraically closed. Now consider the algebraic set $Z(\mathfrak a)$ where $\mathfrak a$ is an ideal of $k[T_1,\ldots, T_n]$ but it is generated ...
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70 views

Divisors of degree $2g-2$ on a hyperelliptic curve of genus $g$

Suppose I have a divisor $D$ of degree $2g-2$ on a hyperelliptic curve of genus $g$. Then I can prove that either a) $K_C\otimes\mathcal{O}(-D)=\mathcal{O}_C$, that is $K_C\cong \mathcal{O}(D)$, or ...
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64 views

Gluing together functions on a closed subvariety

I'm trying to get an intuition for what sheafification does. I came across a passage from Perrin's algebraic geometry book about closed subvarieties. If says that if X is an algebraic variety and Y ...
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1answer
62 views

Cubic hypersurfaces through 5 generic lines in $\mathbb{P}^3$

Consider 5 generic lines $l_1, \dots, l_5 \subset \mathbb{P}^3$ (in particular, they do not intersect). Denote by $Z$ their union. $$\dim H^0 \big( \mathbb{P}^3, \mathcal{O}_{\mathbb{P}^3} (3) \big) = ...
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54 views

Induced scheme structure on an irreducible component?

Suppose that $X$ is a non-reduced scheme of finite type over a field, with multiple irreducible components $X_1,\ldots,X_n$, possibly intersecting each other. Is there a natural scheme structure on ...
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50 views

Definition of a morphism of locally ringed spaces

Let $(X, \mathcal O_X), (Y, \mathcal O_Y)$ be locally ringed spaces. A morphism of ringed spaces is defined to be a pair $(f,f^{\#}):(X, \mathcal O_X) \rightarrow (Y, \mathcal O_Y)$, where $f:X ...
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1answer
51 views

partial solving of ellipse from 5 points

From 5 points on an ellipse I can get the ellipse characteristics (center, radii, angle) by solving a $5\times5$ system (the ellipse equation applied on each point). But this is costly when called ...
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1answer
77 views

Exact sequence of sheaves if and only if exact on the stalks

This is a follow up question to something I asked earlier: What does it mean for a sequence of sheaves to be exact Let $F, G, H$ be sheaves on a topological space $X$, and let $$F ...
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1answer
207 views

Orbits of $SL(3, \mathbb{C})/B$

Let $B= \Bigg\{\begin{bmatrix} * & *&* \\ 0 & *&*\\ 0&0&* \end{bmatrix} \Bigg\}< SL(3,\mathbb C)$. What is $SL(3,\mathbb C)/B$? Do we use these facts: Borel fixed ...
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176 views

Can a conic over $\mathbb{Q}$ with no $\mathbb{Q}$-points have a point of degree 3?

Let $C$ be the conic in $\mathbb{P}^2$ given by $ax^2 + by^2 + cz^2 = 0$ with $a,b,c\in\mathbb{Q}$. (every genus 0 curve over $\mathbb{Q}$ can be given this way right?) Suppose $C$ has no rational ...
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119 views

Galois invariants of the Tate module of an elliptic curve over a number field

Let $K$ be a number field, $E$ be an elliptic curve over $K$, $l \neq p$ be two different prime numbers and $v$ be a place of $K$ above $l$. I am trying to understand the proof of proposition I.6.7 ...
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60 views

Can a bidegree $(3,4)$ curve be embedded in plane?

Suppose $C$ is a curve on $\mathbf{P}^1\times\mathbf{P}^1$ of bidegree $(3,4)$, why such a curve cannot be embedded as a curve in $\mathbf{P}^2$?
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87 views

Smooth Fano Polytopes and Hypersurfaces

This is a rather extended question, so I will try to make it as compact and readable as possible. I am trying to practice with the Macaulay2 software, in particular the polyhedra and ...
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64 views

Hartshorne's proof of the exact sequence $\mathbb{Z} \to \operatorname{Cl} X \to \operatorname{Cl} U \to 0$

Hartshorne, Algebraic Geometry, Proposition II.6.5 reads (in part): Let $X$ satisfy (*), let $Z$ be a proper closed subset of $X$, and let $U = X \setminus Z$. Then: [...] (c) if $Z$ is ...
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78 views

A non-singular quotient of $\mathbb{A}^n$ by a cyclic group is isomorphic to $\mathbb{A}^n$

Let $G$ be a cyclic group acting linearly on $X := \mathbb{A}^n$. If we assume that the quotient $Y:=X/G$ is non-singular, does it follow that $Y \simeq \mathbb{A}^n$? If so, is it necessary to assume ...
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66 views

Computing the sheaf of 1-forms on a toric variety

Consider projective space $P^{2}$ and its corresponding fan. We have the affine opens defined by $U_{\sigma_{0}} = Spec(\mathbb{C}[x,y])$, $U_{\sigma_{1}} = Spec(\mathbb{C}[x^{-1},x^{-1}y])$ and ...
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2answers
187 views

Geometric Intuition Behind Blowing Up a Cusp on a Plane Curve?

I'm reading Hartshorne AG V.3 on monoidal transformations and embedded resolutions. I understand one sort of intuition behind blowing up a point on a surface (or more generally a subvariety of a ...
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1answer
167 views

Proving that certain incidence correspondence is a projective variety.

Let $M$ be the projective space of nonzero $m\times n$ matrices up to scalars (in $\mathbb{K}$). In Joe Harris' Algebraic Geometry: A first course, in order to find the dimension of $M_{k}=\{A\in ...
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32 views

A set is not semialgebraic

A subset $A$ of $\mathbb R^n$ is called semi-algebraic if it can be represented as a finite union of sets of the form \begin{equation*} \{x\in \mathbb R^n\; |\; p_i(x)=0, q_i(x)<0\; \mbox{for all ...
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1answer
80 views

Separated scheme stable under base extension.

Given a separated scheme morphism $X\to Y$, and a morphism $Z \to Y$, Hartshorne proves that the extension $X\times_YZ \to Z$ is also separated, as long as the schemes involved are Noetherian. The ...
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1answer
81 views

Product of schemes and ideal sheaves

Let $X \subset \mathbb{P}^n$ and $Y \subset \mathbb{P}^m$ be projective schemes over $\mathbb{C}$. Then, 1) Is the structure sheaf of $X \times_{\mathbb{C}} Y$ isomorphic to $\mathcal{O}_X ...
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73 views

Does there have to be a point on elliptic curve over $\mathbb{C}(t)$

Let $E$ be an elliptic curve over $\mathbb{C} (t)$ (rational functions). I require $E$ to be defined by the following equation. $$ y^2 = x^3 + A x + B$$ Where $A, B \in \mathbb{C} (t)$. Question: ...
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171 views

Fibres of the base change of a scheme

I am trying to gain a better understanding of the notion of fibre products of schemes. Two major applications that I've began to study are: 1) Making an $S$-scheme $X$ into an $S'$-scheme via a ...
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183 views

What is a geometric interpretation of regular sequences in various instances?

This question arose from my attempts to understand the inclusion Regular $\subset$ Complete Intersection $\subset$ Gorenstein $\subset$ Cohen Macaulay There are many related questions here and in ...
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1answer
155 views

Stiefel-Whitney class of complex projective spaces

Let $T\mathbb{C}P^m$ be the tangent bundle of complex projective space. What is the total Stiefel-Whitney class $w(T\mathbb{C}P^m)$? Let $a_m$ be the maximal integer such that the $a_m$-th dual ...
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1answer
139 views

A cubic hypersurface in $\mathbb{P}^{4}$ that passes through $7$ points in general position with multiplicity $2$.

I am reading Rick Miranda's "Linear systems of plane curves". A cubic hypersurface in $\mathbb{P}^{4}$ that passes through $7$ points in general position with multiplicity $2$ is not expected to ...
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1answer
64 views

Characters of group scheme represented by Cartier dual

For a commutative group scheme $\pi \colon G \to S$ finite locally free over a base scheme $S$ we let $A := \pi_* \mathcal{O}_G$ and $A^* = \mathcal{Hom}_\mathcal{O_S}(A, \mathcal{O}_S)$. Then $A^*$ ...
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212 views

Trouble with Vakil's FOAG exercise 11.3.C

I'm having trouble with the exercise in the title, even with part (a), which asks to prove that if $X$ is a closed subset of $\mathbb{P}^n_k$ of dimension at least 1 and $H$ is a non-empty ...
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124 views

Moore space, induced map in homology

Let $A$ be a finitely generated abelian group and $n$ a positive integer. I have built a connected space $M(A,n)$ such that all its reduced homology groups are zero but the i-th reduced homology group ...
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1answer
115 views

Prove that a presheaf is a sheaf

Let $X$ be a variety. Show that if $X$ is irreducible, then the constant abelian presheaf $\mathcal{F}$ with $\mathcal{F}(U)=\mathbb{Z}$ for every nonempty open subset $U\subseteq X$ and ...
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149 views

direct and inverse images of sheaves and some canonical morphisms

Consider a continuous map $f\colon X\to Y$ between topological spaces. Let $\mathcal F$ be a sheaf on $X$ and $\mathcal G$ a sheaf on $Y$ (let's say of abelian groups). There exists canonical ...