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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12 views

Is there a characterization of contractible hypersurfaces in $\mathbb{C}^2$.

Let $V$ be an irreducible, algebraic hypersurface in $\mathbb{C}^2$ which is contractible as a topological space. I would like to know the algebraic characterization of such objects. For example, ...
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1answer
31 views

Variety that is affine and projective is a finite number of points

I was trying to proof the following without any luck. I would appreciate good hints. A projective variety that is isomorphic to an affine variety is a finite number of points.
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1answer
26 views

suggestion for a book on algebraic curves

I don't know if i'm asking in the wrong place, but I'm studying the book Algebraic curves of Fulton, and having some problems understanding the final chapters (6,7 and 8). I've found a good help in ...
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1answer
28 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 ...
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28 views

Cohomology of affine plane with double origin

How to calculate cohomology $H^1(X,O_X)$,$H^2(X,O_X)$ $H^1(X,O_X^*)$ of affine plane with double origin $X=\mathbb{A}^2\cup_{\mathbb{A}^2-\{0\}}\mathbb{A}^2$? To use Cech cohomology, I cannnot find a ...
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2answers
27 views

Restriction maps in an integral scheme are injective

Suppose $X$ is an integral scheme. I would like to show that the restriction maps $res_{U,V} : O_X(U) \rightarrow O_X(V)$ is an inclusion so long as $V$ is not empty. I was wondering if someone could ...
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0answers
17 views

What are some elementary books which discuss projective lines on surfaces with examples?

I have the books: W. H. Blythe, On models of cubic surfaces (1905) and A. Henderson, The twenty-seven lines upon the cubic surface, and a couple more modern algebraic geometry books including I. R. ...
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1answer
19 views

Show that $\mathbf{I}(\mathbf{V}(x^n, y^m)) = \langle x, y \rangle$.

Show that $\mathbf{I}(\mathbf{V}(x^n, y^m)) = \langle x, y \rangle$. Where $\mathbf{I}$ is the ideal, and $\mathbf{V}$ is the affine variety. I'm not sure how to even begin on this one. I know ...
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0answers
39 views

immersions and finite morphisms

I have the following question: Let $X \subset \mathbb A^n$ be an affine variety. Prove that the immersion $i\colon X \hookrightarrow \mathbb A^n$ is a finite morphism. I know that the ...
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0answers
24 views

What are some applications of parametrization of curves and surfaces?

I know that we can find all elements of a quadratic field with norm 1 by rational parametrization of conics, it can be used to show that some Diophantine equations are not so easy to solve, and that ...
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1answer
35 views

Why do intersection of two quadratic forms implies elliptic curve?

Let $k$ be a field and $S=k[T_0,T_1,T_2,T_3]$ and $f,g\in S_2$ two pairwise irreducible quadratic forms. How can I show that the intersection $X\subset \mathbb P_k^3$ of second degree surfaces ...
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2answers
164 views

Artin Chapter 11, Exercise 9.12, polynomials without common zeroes [on hold]

How do I show that the three polynomials $f_1 = t^2 + x^2 - 2$, $f_2 = tx - 1$, $f_3 = t^3 + 5tx^3 + 1$ generate the unit ideal in $\mathbb{C}[t, x]$? Artin mentions two approaches: by showing that ...
6
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1answer
100 views

How can a finite graph be viewed as a discrete analogue of a Riemann surface?

In the paper "Riemann–Roch and Abel–Jacobi theory on a finite graph" by Baker and Norine, the first line of the abstract states: "It is well known that a finite graph can be viewed, in many respects, ...
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2answers
67 views

Show that if $f \in \mathbb{C} \left[x_1, \dots, x_n \right]$ vanishes at every point of $\mathbb{Z}^n$, then $f$ is the zero polynomial.

I am working on a problem from Ideas, Varieties, and Algorithms: Show that if $f \in \mathbb{C} \left[x_1, \dots, x_n \right]$ vanishes at every point of $\mathbb{Z}^n$, then $f$ is the zero ...
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4answers
411 views

Why there is much interest in the study of $\operatorname{Gal}\left(\overline{\mathbb Q}/\mathbb Q\right)$?

Let's start for a simple quote from wikipedia: "No direct description is known for the absolute Galois group of the rational numbers. In this case, it follows from Belyi's theorem that the ...
3
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2answers
92 views

Why Zariski topology is not Hausdorff

I am reading the book about Algebraic geometry. I am confused about the following two things the book mentioned: Zariski topology is 1. different from the topology studied in real and complex ...
3
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1answer
45 views

Calculating canonical divisor in product of projective spaces.

Let $X$ be an intersection of two divisors of bidegree $(a,b)$ and $(c,d)$ in $\mathbb{P^2}\times \mathbb{P^2}$. Then how can I find the canonical divisor $K_X$? I'm asking because I have no ...
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0answers
25 views

Sparse & Dense Polynomials

I've been reading up on Bernstein's theorem for an algebraic geometry assignment and I've come across the terms "dense" and "sparse" in relation to the polynomials. However, I have been unable to find ...
2
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1answer
25 views

Showing a set is an affine variety

I am trying to work through Hartshorne's book and while working through one of the exercises, I need to show the following: Let $k$ be an algebraically closed field. Let $Y \subseteq A^3$ be the set ...
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0answers
53 views

How to Distinguish Between Base-points in Blowups?

As an example consider the resolution of the base-point (via blowups) of the family of curves in $\mathbb{C}^2$ defined by $f(x,y)=4x^3-ax-b-y^2=0$, where $a$ is a fixed constant and $b$ is a free ...
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1answer
265 views

Working out the value of $x$ on two triangle with the same area in the form $a+\sqrt b$

Here are two triangles T1 and T2. The lengths of the sides are in centimeters. The area of triangle T1 is equal to the area of triangle T2. Work out the value of x, giving your answer in the form ...
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1answer
57 views

Regular Local Ring

Let $Y$ be an affine variety in $\mathbb{A}^n_k$ and $\mathfrak{i}$ its corresponding ideal. We use the notation $A(Y) = k[x_1,...,x_n]/\mathfrak{i}$ for the coordinate ring of $Y$. Pick a point $p\in ...
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1answer
31 views

Basic question related to sheaf of a scheme

Suppose I have a scheme $X$. And some non-empty open set $U \subseteq X$. Does it then follow that $O_X(U)$ is not the trivial $0$-ring by any chance?
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1answer
64 views

Does there exist a regular map $\mathbb{A}^1\to\mathbb{P}^1$ which is surjective?

Suppose $\mathbb{A}^1$ and $\mathbb{P}^1$ are affine space and projective space. I'm not sure if it matters, but I don't mind if we assume that we're working over algebraically closed fields. I'm ...
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1answer
44 views

Why schemes are $(X,\mathcal O_X)$ rather than $(\mathcal O_X,X)$ or $\{X,\mathcal O_X\}$

Is there a reason why schemes are ordered pairs $(X,\mathcal O_X)$ rather than for example $(\mathcal O_X,X)$ or $\{X,\mathcal O_X\}$?
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0answers
22 views

Closed subset of a affine linear group [duplicate]

Let $G\subseteq GL_n(\mathbb{C})$ a Zariski-closed linear subgroup and $X\subseteq G$ closed with $X*X\subseteq X$ and $e \in X$. Then $X$ is a subroup. I am not sure how to start here. I know that ...
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1answer
34 views

local intersection multiplicity

I am reading kenji uneoگس book on algebraic geometry 1. I don't understand how to compute the local intersection multiplicity. I would appreciate if you can show me how to compute it for the next two ...
11
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2answers
918 views

What does the Hodge conjecture mean?

I read from the Internet that according to the Hodge conjecture, a certain harmonic differential form in a projective, non-singular algebraic variety is a rational linear combination of the cohomology ...
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0answers
21 views

Decomposition of abelian varieties up to isogeny

Let $A_1,A_2,B_1,B_2$ be simple abelian varieties over a number field $k$. Suppose that $A_1\times A_2$ is $k$-isogenous to $B_1\times B_2$. Can we deduce that (up to reordering the factors) $A_1$ is ...
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2answers
38 views

pullback of twisting sheaf

Let $[k]: \mathbf{P}^n \to \mathbf{P}^n, [x_0:\ldots:x_n] \mapsto [x_0^k:\ldots:x_n^k]$ be a morphism. (Why) do we have $[k]^*\mathcal{O}_{\mathbf{P}^n}(1) \cong \mathcal{O}_{\mathbf{P}^n}(k)$?
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0answers
40 views

On a method to compute dimension of moduli space of Riemann surfaces

Consider genus $g$ Riemann surfaces, and thier moduli space $\mathcal M_g$. To determine dimension of $T\mathcal M_g$, start with a complex structure, which in some coordinates can be written ...
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1answer
33 views

Quillen's K-theory and ring homomorphisms

I am a beginner in algebraic K-theory and I want to make sure that I understand the following correctly: Let $f:A \to A'$ be an isomorphism of commutative rings. Denote by $\mathcal{P}(A)$ (resp. ...
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0answers
22 views

In P^n(projection of C^n+1) is a variety isomorphic to P^1 irreducible?

In P^n(projection of C^n+1) is a variety isomorphic to P^1 irreducible? I think not because is the union of a line and a point at infinity
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1answer
39 views

Meromorphic function with bounded order of zeros and poles

The following problem has been bothering me for a long time; Let $X$ be a compact Riemann surface of genus $g$. Is there a non-zero meromorphic function on $X$ with all of its poles and zeros have ...
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0answers
35 views

Cartier divisors of schemes

In his notes, Ravi Vakil only defines the notion of an effective Cartier divisor. Furthermore, the Wikipedia page only defines the notion of an effective Cartier divisor for a general scheme. ...
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0answers
15 views

What does the equation $\tau \tau^* = \sigma^* \sigma$ represent in the ADHM construction of vector bundles?

I'm looking at the explicit construction of vector bundles with Anti-Self-Dual (ASD) connections on them via the ADHM construction. At the heart of this is the complex $$ V ...
2
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0answers
27 views

Objects corresponding to Higher forms

If $Q$ is a quadratic form, then we know there exists matrix $A$ such that $Q=xAx'$ and $Q$ can be expressed as weighted sum of eigenvalues of $A$. If $H$ is a higher order form, then is there an ...
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0answers
87 views

Why are period integrals naïve periods?

Apologies for the long question. I recall the definition of a (naïve) period according to Kontsevitch and Zagier [KS]: A (naïve) period is a complex number whose real and imaginary parts are ...
7
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1answer
76 views

Determinant bundle of a tensor product

Let $X$ be a ringed space (for example, a scheme or a manifold). If $V$ is a locally free $\mathcal{O}_X$-module of rank $n$, then $\mathrm{det}(V) := \Lambda^n V$ is a locally free ...
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1answer
68 views

Castelnuovo-Mumford regularity of Cohen-Macaulay modules

Let $S=K[X_1,\ldots,X_n]$ and $M$ be a Cohen-Macaulay $S$-module. This equality holds $$ \operatorname{reg}(M)=\dim(M)+\max\{i\in\mathbb{Z}\colon P_{M}(i)\neq H(M,i)\}. $$ It's been proved in ...
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1answer
26 views

twists of unipotent algebraic groups

Let $U$ be a unipotent linear algebraic group over some field $k$ with char$k$=0. Let $U'$ be a linear algebraic group over $k$ such that $U'_{\bar{k}} = U_{\bar{k}}$ (ie $U'$ is a $\bar{k}/k$-twist ...
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1answer
141 views
+150

Flexes of cubic curve

Which are the flexes of the cubic curve of Fermat $$x^3+y^3+z^3=0$$ at $\mathbb{P}^2(\mathbb{C})$ ? Could you give me a hint how we could find the flexes? Do we have to use maybe the following ...
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1answer
262 views

Good references for stacks

I have seen stacks come up in various settings recently. I understand that, at least heuristically, they are some sort of generalization of a scheme, but I don't actually know anything about them. ...
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1answer
56 views

Simple question about the traslation from french to english

Goodmorning. I'm reading an article by Arnaud Beauville talking about the surfaces of general type. I've found this term and i'm not sure about the translation: What does it mean a " pinceau de ...
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45 views

Book/lecture notes on algebraic curves

Although there surely is plenty of references on MSE about algebraic curves, my need are very specific and so I will open this topic anyway. I follow this year a course on (hyper)elliptic curves ...
2
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1answer
57 views

9 missing lines on a specific smooth cubic surface

Let $\Gamma (x,y,z) = 27 x^3 + 243 x^2 y+324 x y^2 + 189 y^3 +27 x^2 z + 27 x y z - 27 y^2 z + z^3$. $S: \Gamma (x,y,z) = 27 $ is a smooth cubic surface. Consider lines of the form $x = x_0 + p s$, $y ...
2
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1answer
42 views

Blow Up of a Variety

Let $Y$ be an affine variety of $\mathbb{A}^n$ and it contains $0$. Think of $\mathbb{A}^n \times \mathbb{P}^{n-1}$ as a quasi-projective variety. Define a closed subset of $\mathbb{A}^n \times ...
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1answer
55 views

Classifying complex conics up to isomorphism as quotient rings of $\mathbb{C}[x,y]$

This is a continuation of the question I asked here. The problem is now: Let $Q = ax^2 + bxy + cy^2 + dx + ey + f \in \mathbb{C}[x,y]$ be a general quadratic polynomial, that is, $a,b,c \not= 0$. ...
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1answer
69 views
+50

The intersection points are collinear

It is given a hexagon inscribed in a conic section. I want to prove that the pairs of opposite site intersect at three points of the projective plane that are collinear. How could we do this? ...
4
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1answer
53 views

Definition of $\mathbb{A}^n_S$ by glueing

In Eisenbud and Haris (Geometry of schemes I.2.4), if $S=\cup_\alpha U_\alpha$ with the $U_\alpha$ affines, to define $\mathbb{A}^n_S$ one take $X=\cup_\alpha \mathbb{A}^n_{U_\alpha}$ with ...