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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Newton's Investigation of Cubics: Generalization?

I recently read about Newton's investigation of cubic curves, and how, like for quadratic curves we can classify them into parabolas, ellipses and hyperbolas, Newton was able to classify cubic curves ...
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15 views

“Pseudomanifold with no singularities”

An $n$-dimensional (closed) pseudomanifold is a finite simplicial complex $X$ such that (i) every simplex is a face of an $n$-simplex (ii) every $(n-1)$-simplex is a face of exactly two ...
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1answer
57 views

Computing Images of Varieties

Somehow, this problem has been coming up a lot lately in different guises, which I'm taking as a sign that I ought to stop avoiding computational algebraic geometry. I could probably dig this up in ...
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0answers
41 views

Upper semicontinuity of fibre dimension on the target

This is Vakil 18.1.C. Suppose $\pi : X \to Y$ is a projective morphism where $Y$ is locally Noetherian (or more generally $\mathcal{O}_Y$ is coherent over itself). Show that $\{y \in Y : \dim ...
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1answer
49 views

A question regarding evaluating a function of a scheme

I am learning about schemes reading Ravi Vakil's notes. On page 136, he says "For example, consider the scheme $\mathbb{A}_k^2 = Spec \ k[x,y]$, where $k$ is a field of characteristic not $2$. Then ...
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1answer
51 views

References about algebraic geometry

My question is very simple. I'm studying a course telling about algebraic surfaces but i think that i need some knowledge about basic algebraic geometry. Do you have some suggestions?
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37 views

Comparing two definitions of determinant of coherent sheaves

Let $f:X \to S$ be a smooth, projective morphism of $k$-schemes for some field $k$. Let $\mathcal{F}$ be a coherent sheaf on $X$ flat over $S$. We know (by Proposition $2.1.10$ of Huybrechts-Lehn, ...
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2answers
44 views

How would I derive the equations of the family of lines on a hyperbolic paraboloid?

My textbook writes out what the equations of the two one-parameter families of lines that lie on a hyperbolic paraboloid surface are, but I am having trouble figuring out how these would have been ...
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1answer
56 views

$I(Y) = \{ p(x,y,z) \in k[x,y,z] \mid p (t,t^2,t^3) = 0, \forall t \in k \}$ is prime

I've been working on the following problem from Hartshorne: Let $Y\subseteq \mathbb{ A }^3 $ be the set $Y = \{(t,t^2 , t^3) \mid t \in k \}$. Show that $Y$ is an affine variety of dimension $1$. To ...
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0answers
15 views

QR decomposition, borel groups and generalizations

Then every matrix $M$ in $M_{m\times m} (\mathbb{C})$ can be written in the form: $QR=M$, where $Q$ is unitary and $R$ is upper-triangular. My question is simple, does this generalize in the ...
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1answer
64 views

determining the blowup of $Y^2=X^3+X^2$ at the origin

Let $C = \{(x,y) \in \mathbb{A}^2_k : y^2 = x^3+x^2\}$ as an affine variety over some algebraically closed field $k$. From the real picture we see that this curve is self-intersecting at the origin. I ...
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1answer
26 views

what is the definition of Section

Giving a surjective morphism $\phi : S \rightarrow C$ from e complex algebraic projective surface to a projective curve i've found the term section of the morphism $\phi$ but i don't have some ...
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1answer
27 views

What about the reducible fibers of a surjective morphism?

If i have a surjective morphism $\phi :S \rightarrow C$, where $S$ is a smooth complex algebraic projective surface and $C$ a smooth projective curve, what can i say about the number of the fibers of ...
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0answers
23 views

Generators of a function field and classification

in the book "Introduction to Compact Riemann Surfaces and Dessins d'Enfants" the autor classify all the compact Riemann surface of genus 3 associated to the curves on form $F(X,Y) = 0$ where $F(X,Y) = ...
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1answer
52 views

Algebraic Curves: Valuation at a point

I would like to understand the notion of valuation on the local ring of a curve at a point. In the Book The Arithmetic of Elliptic Curves in chapter 2, Example 1.3 $$V:\ Y^{2}=X^{3}+X$$ I don't ...
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0answers
31 views

Condition for the generic vector bundle to be globally generated

In the paper "Gwena, Teixidor - Maps between moduli spaces of vector bundles and the base locus of the theta divisor", it is stated without proof that a $general$ vector bundle on a curve of rank $r$ ...
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1answer
32 views

Relative center of relative group scheme

This might be an easy question. Let $p: X \rightarrow S$ be a relative group scheme. In particular the fibers are group schemes. I want to know if there are constructions like the ``relative ...
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1answer
66 views

Prove that a subset is a finitely generated subring

Consider $\mathbb{A}^2$ with $\rho : (x, y) \mapsto (-x, -y)$. Can anyone help me prove that $S = \{f \in \mathbb{C}[x, y] : f \circ \rho = f\}$ is a finitely generated subring? Also, can $S$ be ...
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1answer
60 views

Reading Griffiths Harris: Quick question

Why is a meromorphic section without zeros and poles on a compact Riemann surface necessarily a constant? Thank you very much.
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0answers
25 views

Solving a simple Distance Geometry problem

I'm trying to solve the following problem: Given the absolute positions of four points in 3D space, and the distances from these four points to a fifth point, find the position of the fifth point. ...
0
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1answer
26 views

Existence of a rational function on a nonsingular curve with a simple zero at P and order 0 at Q

Let $X$ be a nonsingular curve over an algebraically closed field $k$ (by curve over $k$, I mean an integral separated scheme of finite type over $k$ which has dimension 1). Let $P$ and $Q$ be ...
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1answer
76 views

Atiyah-Macdonald 5.2

Exercise 5.2 in Atiyah-Macdonald asks to show the following: "Let $A$ be a subring of a ring $B$ such that $B$ is integral over $A$, and let $f: A \to \Omega$ be a homomorphism of $A$ into an ...
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0answers
65 views

Enriques Noether Theorem

I'm reading the proof of the Enriques-Noether theorem (Beauville A. page 25-26-27). here the stament: Let $p:S \rightarrow C $ a surjective morphism between an algebraic surface $S$ and a smooth ...
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1answer
23 views

Complete the square to change into standard form

Here is the equation: $x^2 + y^2 + 4x - 6y - 3 = 0$ Here are the instructions: Complete the square to change the equation info standard form. Then graph the equation. Because both $y^2$ and ...
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1answer
34 views

Notation in Dolgachev's CAG

On page 497 (page 509 of the pdf) of Dolgachev's CAG (www.math.lsa.umich.edu/~idolga/CAG.pdf) he uses the notation "$\succ_1$". Can someone explain me what that means?
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2answers
78 views

Going-up and going-down theorems: motivation

I am reading about the going-up and going-down theorems in Atiyah & Macdonald's commutative algebra book. I'm wondering if anyone could give me some basic facts/examples to help me understand why ...
3
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1answer
60 views

If $f_1,…,f_{n+1}\in\mathbb{C}[x_1,…,x_n]$, is there a polynomial in the coefficients which vanishes iff the $f_i$ have a common root?

My question is as in the title: Suppose $f_1,...,f_{n+1}\in \mathbb{C}[x_1,....,x_n]$. Is there polynomial $g$ (or a system of polynomials) with variables given by the coefficients of the $f_i$ ...
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0answers
28 views

compute euler characteristic for pull back line bundle of finite map

i want to know if there is any formula to compute the euler characteristic for pull back line bundle of finite morphism, and the same question for blow up one point? Espesically for algebraic ...
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0answers
17 views

Isolated points in the Chow scheme, and rigidity

Let $C$ be a smooth projective curve embedded in a smooth projective variety $X$, for example a threefold. The curve defines a cycle $[C]$, i.e. a point of the Chow scheme $\textrm{Chow}_1(X;d)$ of ...
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1answer
56 views

Are smooth varieties locally isomorphic to the affine space?

A smooth $n$-dimensional manifold is locally isomorphic to $\mathbb{R}^n$. I am wondering if the analogous statement for smooth algebraic varieties is also true. Let $X$ be an $n$-dimensional ...
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24 views

Application of the Castelnuovo's contraction criterion (CCC)

I've seen the castelnuovo's contraction criterion that says: if $E \in S$ is a $(-1)$ curve contained in a complex projective algebraic surface than $E$ is an exceptional curve. Then there is an ...
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44 views

Endomorphisms of constant sheaves on connected spaces

In a paper by Deligne and Lusztig it says An endomorphism of a constant sheaf over a connected base is constant My interpretation of this statement is that given a (non-empty) connected ...
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0answers
40 views

A query on Veronese mapping

The Veronese mapping defined as usual on some $P^n$. Then it is certainly regular. I want to prove that the inverse map to this map is also regular. I have an idea to use projections with ...
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2answers
28 views

minimal prime ideals over the union of two prime ideals

When two subvarieties intersect properly ($X_1\cap X_2$), it should end up with a new subvariety($X_3$=$X_1\cap X_2$). I do not know how to keep track of the intersection operation from the algebraic ...
3
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1answer
50 views

Are any (non-empty) Euclidean open sets dense in the Zariski topology?

It's well known and easy to show that every Zariski open set is dense in the Zariski topology. However I search the web and didn't find an answer to my question, which I believe is true. My ...
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0answers
25 views

When Is A Quot Scheme Reduced?

For my research, I would like to know whether a certain Quot scheme is reduced. Reading the thread How To Tell Whether A Scheme Is Reduced From Its Functor, I was disappointed to find that there's no ...
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1answer
27 views

Extension of coherent sheaf from open subspace.

I'm trying to solve following problem: Let $X$ be a noetherian scheme, $U$ an open subspace of $X$, $\mathcal F \in Qcoh(X), \mathcal G\in Coh(U), \mathcal G \subset \mathcal F|_{U},$ then there ...
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0answers
39 views

A couple of questions on an elliptic fibration $S\to\mathbb P^1$

Let $C_1,C_2\subset \mathbb P^2$ be two plane cubics intersecting in $9$ points $p_1,\dots,p_9$. If $C_i=V(s_i)$, there is a rational map $f:\mathbb P^2\dashrightarrow \mathbb P^1$ defined (away from ...
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2answers
83 views

Eisenbud & Harris's The Geometry of Schemes proof of Prop I-18

There seems to be an error in the proof of Proposition I-18. The first inset equation (line 7) on page 20 only holds in $X_{f_a f_b}$. But it is used on line 15, where it needs to hold in $X_{f_b}$. ...
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0answers
40 views

Quasi-projective variety

I have a simple question about definitions. If X is a quasi-projective variety, then what does it mean for $U_i$ to be a cover by affine varieties? If X lives in projective space, how can it have ...
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1answer
43 views

Curve in an algebraic variety

Let $\lambda_1, \lambda_3, \lambda_3$ be distinct real numbers. Can it be that a curve of the form $$ t \mapsto \gamma(t) := (e^{\lambda_1 t}, e^{\lambda_2 t}, e^{\lambda_3 t}) $$ is contained for all ...
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1answer
42 views

On a bijection between isomorphism Spec $A$ $\rightarrow$ Spec $A'$ and the ring isomorphism $A \rightarrow A'$.

Let $A$ and $A'$ be commutative rings with unity. Suppose I have an isomorphism of affine schemes, say: $\pi:$ Spec $A$ $\rightarrow$ Spec $A'$ and $\alpha: O_{Spec A'} \rightarrow \pi_* O_{Spec A} ...
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1answer
31 views

What to think of $A_{\mathcal{P}}$ in Spec $A$

Suppose I have a commutative ring with unity $A$, $f \in A$ and a prine ideal ${\mathcal{P}}$. I have learned that $A_f$ can be naturally thought of as an open subset of Spec $A$. I was wondering if ...
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1answer
41 views

etale sheafification of a presheaf

Consider the following presheaf on the big etale site of smooth schemes over a field $k$: to every smooth $k$-scheme $U$, associate $$F(U):= \{f: U \to \mathbb A^1_k ~|~ \text{$f$ factors through the ...
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votes
1answer
65 views

The role of valuation rings in algebraic geometry

I am familiar with basic algebraic geometry in the tradition of Hartshorne's book. Discrete valuation rings appear there in the criteria for separatedness/properness, and are used to define the order ...
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0answers
42 views

Preimage of a variety under a morphism

Is the preimage of a variety a variety, or is the preimage of an affine variety an affine variety? More precisely, Let $X,Y$ prevarieties, $\varphi: X \to Y$ a morphism. given a (affine) variety $Z ...
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1answer
76 views

Finding a locally free resolution for the sheaf of ideals of a hilbert scheme's universal family

I really hope the question's title isn't misleading, but unfortunately no better one came to my mind (EDIT: I adjusted the title, but i'm still not happy with it). I'm trying to understand the ...
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1answer
31 views

unirational complex variety has $H^i(X,O_X) = 0$ for i > 0

Let $X/\mathbf{C}$ be a smooth projective connected unirational variety. Why do we then have $H^i(X,O_X) = 0$ for $i > 0$?
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1answer
41 views

Understanding the mechanics of gluing sections of presheaves to obtain sheaves?

Could anyone give me a couple specific examples of how sections of a Presheaf on discrete topology would or could glue together? If I am correct, it depends on the mapping one defines. Now I have ...
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1answer
59 views

morphism of a variety onto affine space with an infinite fiber

There is the following geometric interpretation of Noether's normalization theorem: Let $X$ be an $n$-dimensional affine variety. Then there is a surjective morphism $\varphi : X \to \mathbb{A}^n$ ...