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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Pole set of rational function defined on a variety

The problem: Let $V = V(y^2-x^2(x+1))$, and let $\overline{x}, \overline{y}$ denote the $I(V)$-residues of $x$ and $y$ in the coordinate ring $\Gamma(V)$. Set $z=\overline{y}/\overline{x}$. Find the ...
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220 views

An exact sequence of homology in abelian categories

$\renewcommand{\im}{\mathop{\rm im}}\DeclareMathOperator{\coker}{coker}$Let $A\xrightarrow{f}B\xrightarrow{g}C$ be a complex in an abelian category, I.e. $gf=0$. Let $H:=\coker(\im(f)\to \ker(g)).$ ...
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176 views

Lines on algebraic surfaces and duality

In the following, all varieties will be algebraic over $\mathbb{C}$. I have some general problems with concepts like the "space of lines in $\mathbb{P}^5$", "space of lines on a surface in some ...
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182 views

Embeddings of bundles in projective space.

Consider the projective variety $X = \mathbb{P}^2$, and the line bundle $\mathcal{O}_X(dH)$ where $H$ is a plane and $d \in \mathbb{N}$. Let $L$ be the total space of $\mathcal{O}_X(dH)$. I know how ...
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587 views

Canonical form of conic section

I have $x^2+2xy-2y^2+x-4y=0$ and I have to find its canonical form, but I'm a little confused.. I'd like to understand very well what I have to do.. Can you help me, please? Thanks!
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1answer
159 views

Is morphism between curves projective?

Suppose $X$ is a smooth, projective curve, $Y$ is an arbitrary curve(may be singular), and both curves are over an algebraically closed field $k$ with character 0. Let $f: X \to Y$ be a morphism ...
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1answer
294 views

Roots of rational equation with multiple variables?

Let's say we have a rational polynomial in $k$ variables. We are only interested in rational solutions. If $k = 1$, name the variables ${x}$, if $k = 2$, name them ${x,y}$. For $k = 1$, it can be ...
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1answer
102 views

Corollary 11.13 in Harris' Algebraic Geometry, a first course

I'm confused about corollary 11.13 (see e.g. google books) in the afforementioned book, namely its second half (the dimension formula). If we take $X$ to be the disjoint union of a point and a line, ...
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326 views

Question about integral closures and localizations

Suppose $A$ is an integral domain with integral closure $\overline{A}$ (inside its fraction field), $\mathfrak{q}$ is a prime ideal of $A$, and $\mathfrak{P}_1,\ldots,\mathfrak{P}_k$ are the prime ...
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106 views

Show that if the curve $y^2 = p(x)$ has a double point, then it must be of the form $(r,0)$ where $r$ is a double root of $p(x)$.

Let $p(x) = ax^3 + bx^2 + cx + d$ where $a,b,c,d \in\mathbb{R}$. Show that if the curve $y^2 = p(x)$ has a double point, then it must be of the form $(r,0)$ where $r$ is a double root of $p(x)$. ...
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165 views

Exact sequence in Beauville's “Complex Algebraic Surfaces”

On page 3 of Beauville's book (Lemma I.5) he takes two curves $C$ and $C'$ in a surface $S$ an takes global sections $s\in H^0(S,\mathcal{O}_S(C))$ and $s'\in H^0(S,\mathcal{O}_S(C'))$. In a recent ...
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104 views

$\mathrm{Aut}(\mathbb P)$ is isomorphic to $\mathrm{ Aut} (k(t) )$

$\def\Aut{\mathrm{Aut}}$I want to prove that the automorphism group of $\mathbb P^1$ it's isomorphic with the Moebius transformation with coefficients over the obvious field. I proved that the ...
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1answer
135 views

Prove/disprove complex affine curves are isomorphic

Define: $X_{1}:=\{(x,y,z) \in \mathbb{C}^{3}: x^{3}-y^{5}=0\}$ $X_{2}:=\{(x,y,z) \in \mathbb{C}^{3}: z=0,x=y^{2}\} \cup \{(x,y,z) \in \mathbb{C}^{3}: x=0,y=z^{2}\}$. $X_{3}:=\{(x,y,z) \in ...
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78 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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62 views

How to show that a variety cannot be described as the intersection of less than three quadrics

a trivial question perhaps. Given the projective variety defined by the condition $$ rank \left( \begin{array}{lll} x_0 & x_1 & x_2 \\ x_1 & x_2 & x_3 \end{array} \right) \leq ...
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1answer
67 views

a question on deforming a ring

I'm trying to learn about deformation theory. Consider $k[x,y]/\left< y^2 - x^2\right>$. To deform $k[x,y]/\left< y^2 - x^2\right>$ to make it look like $k[x,y]/\left< ...
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243 views

Counting multiplicities and Bezout's theorem

What is the method to count multiplicities of intersection? for example suppose we have the projective line $x=0$ in $\mathbb{P}^{2}$ and the curve $V(z^{2}y^{2}-x^{4}) \subseteq \mathbb{P}^{2}$. ...
3
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1answer
179 views

Explicit example of a toric flip

I am looking for a toy example of a flip between toric projective 3-folds. More precisely, I would like to see their defining fans (or polytopes). Does anyone know where I can find something like ...
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1answer
319 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
57 views

Morphism $f : \mathbb C^n \to \mathbb C^m$ whose image is not algebraic?

How can I construct a polynomial function $f : \mathbb C^n \to \mathbb C^m$ whose image is not algebraic? I can't really get anywhere here. Any hints would be greatly appreciated. Thanks
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1answer
134 views

Ample line bundles and dimension of a proper variety

Let $Y$ be a proper variety over a field $k$. Suppose that $\mathcal{L}$ is an ample line bundle on Y and suppose that $\mathcal{L}$ is isomorphic to the trivial bundle. What can I conclude about the ...
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262 views

Apparent contradiction using Mayer-Vietoris for Sheaf cohomology

Trying to solve Exercise 2.7 b) of Chapter III of Hartshorne's Algebraic Geometry I got stucked about an apparent contradiction. The exercise asks to prove that $H^1(S^1, \mathcal{R})=0$, where $S^1$ ...
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120 views

when the curve $\mathbb{r=a\sin(b\theta)}$ is algebraic?

A need to show that the curve given in polar equation $\mathbb{r=a\sin(b\theta)}$ is an algebraic curve if $b=\frac{m}{n}$, $m,n\in \mathbb{N}^{*}$ and $(m,n)=1$. Also I am supposed to find the ...
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151 views

Vanishing of $ H^1(\mathcal{M})$ implies vanishing of $H^1(U\otimes\mathcal{M}) $ on a curve.

Let $C$ be a smooth projective curve of genus $g\geq 1$ over an algebraically closed field. Let $\mathcal{M}$ be a line bundle with $deg \mathcal{M}\geq 2g -1$. Let $T$ be torsion and denote by $U$ ...
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171 views

Extension of Vector bundles is a Vector bundle?

I guess this is quite easy, but I don't see a counterexample: let $X$ be a noetherian scheme (maybe with more hypotheses, but I don't think this will change much), then I have the feeling that it is ...
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1answer
102 views

Points in integral separated schemes are determined by their local rings

I am a bit stuck with the following assertion: Let $X$ be a separated integral scheme. Then to every (schematic) point $x \in X$ we can correspond its local ring, and look at it as a subring of the ...
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314 views

Epimorphisms of sheaves of sets

Let $X$ be a topological space, and $F$ and $G$ be two sheaves of sets on $X$. Let $\eta : F \rightarrow G$ be a morphism of sheaves. Then how would you show the following: $\eta$ is an epimorphism ...
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90 views

Does every Jacobian over $\overline{\mathbf{Q}}$ have everywhere good reduction?

Let $J$ be the Jacobian of a smooth projective connected curve of genus $g>1$ over the field $\overline{\mathbf{Q}}$ of algebraic numbers. Does $J$ have everywhere good reduction? I know that ...
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1answer
171 views

What is going on with the map $X \mapsto X^2$

Consider the homomorphism $f : \mathbb{C}[X] \to \mathbb{C}[X]$, $X \mapsto X^2$. It induced a morphism of affine schemes $\operatorname{spec} f : \mathbb{A} \to \mathbb{A}$ which topologically is the ...
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1answer
167 views

$M_n\cong\Gamma(\operatorname{Proj}S.,\widetilde{M(n).})$ for sufficiently large $n$

Let $S.$ be a graded ring, finitely generated by degree 1 elements as a $S_0$-algebra. Let $M.$ be a finitely generated graded $S.$-module. There exists a natrual map ...
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1answer
180 views

Weierstrass Equation and K3 Surfaces

Let $a_{i}(t) \in \mathbb{Z}[t]$. We shall denote these by $a_{i}$. The equation $y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}$ is the affine equation for the Weierstrass form of a ...
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1answer
499 views

Presheaf which is not a sheaf — holomorphic functions which admit a holomorphic square root

I'm thinking about a problem in Ravi Vakil's algebraic geometry notes http://math.stanford.edu/~vakil/216blog/ (Exercise 3.2B) and I'm having trouble with the second part of the exercise as my ...
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1answer
324 views

Divisors of rational functions on curves at singular points

Suppose $C$ is an algebraic curve (which has singular points) over an algebraically closed field $k$, and that $f$ is a rational function on $C$. How does one defines the Weil divisor of $f$? The ...
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247 views

what are good references for learning about vector bundles and their sheaves of sections?

I am a beginner in representation theory and algebraic geometry, so that references giving clear explanations of things like the tautological line bundle on $\mathbb P^n$, its dual, and the associated ...
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1answer
348 views

What is the pullback in the category of commutative algebras?

The pullback is a subset of the cartesian product in the category of commutative rings with unit. What is the pullback in the category of commutative $k$-algebras? Is it the same set as in rings?
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195 views

How to find a bijection between R-valued points of X and local ring homomorphisms?

I am trying to prove the following fact for a homework assignment in algebraic geometry: Let R be a local ring, and X a prescheme. Show that there is a one-one correspondence between R-valued points ...
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282 views

To what extent do the stories on manifolds carry over to schemes?

This is a follow-up (refinement?) of this question. In learning some algebraic topology, I've learned to think of an affine scheme as spec $R$. (I've been told that this is a legitimate use of ...
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2answers
237 views

Flatness of local rings

What do I miss in the following? Let $R$ be a commutative Noetherian ring with unit. A map $f:M\to N$ of $R$-modules is injective/surjective iff the associated map $f_p:M_p\to N_p$ on the ...
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60 views

$\dim (D-P)=\dim (D)-1$

I'm trying to prove this question: Let $D$ be a divisor in $F|K$ such that $\dim (D)\gt 0$ and $0 \neq f\in \mathscr L(D)$. Thus $f\notin \mathscr L(D-P)$ for almost all $P$. Then show that ...
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52 views

Hartshorne Ex III5.7 c)

Suppose $X$ is a reduced proper schemes over a noetherian ring, $X_i$ are its irreducible components,$L$ is a invertible sheaf on $X$. If $L|_{X_i}$ are all ample, how do we show $L$ is ample? If ...
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98 views

sheafification definition?

I came across two different definitions of sheafification and I'm not sure how they are equivalent. One of them is here: About the sheafification Another one is from Tennison's sheaf theory: Given a ...
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66 views

Bijection between solutions of polynomials equations and spectrum of the quotient ring.

I would like to see the connection between the set of solutions of a system of polynomial equations and a spectrum of the quotient polynomial ring. Given $f_1, f_2,\cdots, f_m \in k[x_1,x_2,\cdots, ...
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1answer
55 views

Two questions on the definition of $\mathcal{O}_X(U)$ for an affine scheme $X$.

Let $X=\operatorname{Spec}(A)$ be an affine scheme. Hartshorne defines $$ \mathcal{O}_X(U)=\{s\colon U\to\coprod_{\mathfrak{p}\in U} A_\mathfrak{p} \mid s(\mathfrak{p})\in A_\mathfrak{p} \text{ and } ...
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1answer
39 views

Adjunction counit for sheaves is isomorphism

Let $f\colon X \to S$ be a proper morphism of varieties over $\mathbb{C}$ with $f_* \mathcal{O}_X = \mathcal{O}_S$ and $\mathcal{G}$ be a coherent sheaf on $S$. Then we have a natural morphism ...
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80 views

Why do only fixed points contribute to the Euler characteristic?

Let $G$ be an algebraic group with zero Euler characteristic, acting on a variety $X$ (over $\mathbb C$). I read some time ago that then the Euler characteristic of $X$ can be computed as ...
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1answer
82 views

Torsion sheaves on a curve

This is probably a silly question, but I'm a bit confused. Regarding exercises 6.11 and 6.12 of Chapter II of Hartshorne: Let $X$ be a nonsingular projective curve over an algebraically closed field ...
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1answer
83 views

Why is it called “elliptic” curve?

One of my favourite and most studied algebraic curve is the elliptic curve. But something that I have never asked myself is: Why do they call this nonsingular cubic curve an "elliptic" curve? ...
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1answer
76 views

rank of quadrics

Consider the quadric $xw-yz$ in $\mathbf{P}^3$ (all over $\mathbf{C}$), and the Klein quadric $x_0 x_5+x_1 x_4+x_2 x_3$ in $\mathbf{P}^5$. I want to determine the rank of these quadrics. For the first ...
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1answer
42 views

Degree of ample bundle over projective curve is positive

(From Vakil's notes, Exercise 18.4.K) If $C$ is an integral projective curve over a field $k$, and $\mathscr{L}$ is an ample line bundle on $C$, why is the degree of $\mathscr{L}>0$? If $C$ is ...
3
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1answer
58 views

local parameter on an irreducible affine algebraic curve

On page 14 of Shafarevich's Basic Algebraic Geometry 1, it is stated that for an irreducible affine algebraic curve $X: f(x,y) = 0$, and a nonsingular point $P \in X$, there is a regular function $t$ ...