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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Calculating Hodge numbers by means of locally free resolutions

In this paper the author considers a smooth $3$-fold $X$ in $\Bbb{CP}^6$ with the following locally free resolutions of its structure sheaf and squared ideal sheaf: $$0\to \mathcal O_\Bbb {P^6}(-7) ...
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56 views

Family of quartic surfaces in $\mathbb{P}^3$ that contain a fixed line or conic

Let $V$ be a complex vector space of dimension 4 and $\mathbb{P}^3=\mathbb{P}(V)$. The space of quartic surfaces in $\mathbb{P}^3$ is $\mathbb{P}(\text{Sym}^4(V^*))$ and the dimension of this space is ...
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1answer
37 views

Prove that $I_k \otimes_k \Omega \rightarrow I$ is injective

Let $\Omega$ be an algebraically closed field, $k$ a subfield of $\Omega$, $I$ an ideal of $\Omega[X_1, ... , X_n]$, and $I_k = I \cap k[X_1, ... , X_n]$. Then $I_k$ is an ideal of $k[X_1, ... , ...
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1answer
29 views

Definition of a fan of a polytope

In Fulton's book Introduction to Toric varieties (page 25), he says that: A rational convex polytope $K$ in $N_{\mathbb{R}}$ determines a fan $\Delta$ whose cones are the cones over proper faces ...
3
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1answer
139 views

Proving exactness of the conormal sequence

Problem: Let $\phi \colon A \to B$ be a surjective homomorphism of $R$-algebras with kernel $I$. I want to show that the conormal sequence $$ I/I{}^2 \longrightarrow B \otimes_A \Omega_{A/R} ...
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1answer
28 views

Unicity of a projective transformation determined by 5 points in $CP^3$?

Consider an ordered set of five points $\{p_1, p_2, \dots, p_5\}$ in linear general position in $\mathbb{CP}^3$ and another ordered set of five points $\{q_1, q_2, \dots, q_5\}$, also in linear ...
3
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1answer
32 views

$(A_f)_{g/f^{n_0}}\cong A_{fg}$ (localization with the powers of an element)

I'm working in a problem from Hartshorne Algebraic Geometry. But I need a result from Commutative Algebra. Given a commutative ring $B$ with $1$. For each $b \in B$ define the ring $B_b$ as the ...
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1answer
29 views

Singular plane cubic curve birational to $\mathbb{P}^1$

Is it true that every singular plane cubic curve over an algebraically closed field is birationally equivalent to $\mathbb{P}^1$? I know that such a curve has to have only one singular point and that ...
2
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0answers
46 views

Uniqueness of a projective transformation

Just as there exists a unique projective transformation that takes three points in $\mathbb{CP}^1$ to three other points in $\mathbb{CP}^1$, how many points do I need for the corresponding question in ...
2
votes
1answer
62 views

Divisor on curve of genus $2$

I suffer from lack of concrete examples in Algebraic Geometry, so I will appreciate it if somebody can help me in understanding a bit better this one: Let $\mathcal{C}$ be a genus $2$ curve ...
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38 views

Mapping a curve into projective space

Let $\mathcal{C}$ be a (smooth, complex, projective) genus 2 curve. Take two different points $p,q\in\mathcal{C}$ and let $K$ be the canonical divisor class. I know (by means of Riemann-Roch) that the ...
3
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1answer
33 views

Does there exist a non-quasi-split torus?

In a homework, I was asked to prove that any torus is isomorphic to a quotient of a finitely many product of Weil restrictions $Res_{L/k}\mathbb{G}_m$. While solving this, I got an impression that ...
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1answer
43 views

Resolving a node singularity on a plane curve

So, I am trying to solve Exercise 1.5.6 b) on page 37 in Hartshorne's Algebraic Geometry. For completeness I will include the exercise in my post: If $P$ is a node on a plane curve $Y$, show that ...
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0answers
18 views

Stabilizers of Segre varieties

What, if anything, is known about maps in PGL(V) that preserve Segre varieties? I am specifically interested in linear maps preserving the Segre embeddings of $\mathbb{P}^{15} \times ...
2
votes
1answer
64 views

Extending a morphism of schemes

This question is an exercise 2.4 p.96 from Qing Liu's book "Algebraic Geometry and Arithmetic Curves". Let $X$, $Y$ be schemes over a locally Noetherian scheme $S$, with $Y$ of finite type over $S$. ...
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1answer
143 views

If $p$ is a positive multivariate polynomial, does $1/p$ have polynomial growth?

I wanted to ask a separate question to focus on an elementary issue from my question Does the inverse of a polynomial matrix have polynomial growth?. Let $p : \mathbb{R}^n \to \mathbb{R}$ be a ...
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0answers
8 views

Pre-image of a measure-zero set of a bilinear mapping?

consider a bilinear mapping $f: R^m \times R^k \rightarrow R^n$. What is the condition on the mapping such that the preimage of any measure-zero set is measure zero? Is there anyway to avoid checking ...
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2answers
44 views

What is the permutation representation of $SL_2(\mathbb{F}_p)$, $PSL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$?

It seems that there is a action by which $SL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$ permute the $p^2$ ordered tuples in $\mathbb{F}_p^2$. What is the map from the $2 \times 2$ matrices over ...
2
votes
1answer
45 views

Subscheme of projective space in general position

Let $k$ be a field and let $\mathbb{P}^n(k)$ denote $n$-dimensional projective space over $k$. What is meant by a general linear space in $\mathbb{P}^n(k)$ of codimension $m$, in the language of ...
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46 views

Image of point of codimension one has codimension one?

I'm working on the following exercise from Liu's Algebraic Geometry and Arithmetic Curves: Let $f: X \rightarrow Y$ be a morphism of Noetherian schemes. We suppose that either $f$ is flat or $X,Y$ ...
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1answer
23 views

morphism betweem sheaves that is an isomorphism in local sections of a basis

Let $\{U_{\alpha}:\alpha \in A\}$ be a basis of open sets for the topological space $X$. Let $\mathscr{F},\mathscr{G}$ be sheaves over $X$. Suppose that there exist a morphism $\phi: \mathscr{F} \to ...
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2answers
56 views

Local ring inside a function field of transcendence degree one

Let $K$ be a function field of transcendence degree 1 over a base field $k$. Let $(R,\mathfrak{m}) \subseteq K$ be a local ring that is not a field. Suppose $S,T$ are DVR's of $K$ which dominate $R$ ...
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1answer
44 views

dimension of quotient by algebraically independent elements

Let $f_1,\dots,f_s$ be algebraically independent polynomials of $A:=k[x_1,\dots,x_n]$, $s \le n$. Recall that algebraically independent means that there is no non-zero polynomial $g \in ...
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1answer
36 views

Definition of an algebraic set being “defined over $k$” in terms of tensor product

Let $\Omega$ be an algebraically closed field, $I$ a radical ideal, and $k$ a subfield of $\Omega$. If $I = \mathcal I(A)$ for some closed set $A \subseteq \Omega^n$, then $I_k := I \cap k[X_1, ... , ...
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1answer
185 views

Mordell-Weil rank in elliptic surfaces

Suppose that an elliptic smooth K3 surface $X$ defined over a number field $k$ has arithmetic Picard rank $r$ and assume that it is equipped with a $k$ fibration over $\mathbb{P}^1$ that has a section ...
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1answer
47 views

Extending regular function on normal variety from a subvariety of codimension 2

In his book "Commutative Algebra with a View Toward Algebraic Geometry" Eisenbud proves the Corollary 11.4 which states the following If $R$ is a normal Noetherian domain, then $R$ is the ...
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1answer
56 views
+50

How to show that $\mathfrak{m}/\mathfrak{m}^{2}\rightarrow\mathfrak{m}A_{\mathfrak{m}}/\mathfrak{m}^{2}A_{\mathfrak{m}}$ is an isomorphism?

Let $A$ be a ring and $\frak{m}$ a maximal ideal of $A$. Let $\kappa$ be the field $A/\frak{m}$. How to show that the $\kappa$-linear natural map $$ ...
0
votes
1answer
42 views

dimension inequality for graded versus non-graded polynomial rings

Let $A=k[x_1,\dots,x_n]$ be a polynomial ring over an algebraically closed field $k$. Let $I$ be an ideal of $A$ and $f$ some element of $A$. Then the Krull dimension does not necessarily satisfy the ...
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3answers
78 views

Convert quadratic bezier curve to parabola

A quadratic Bézier curve is a segment of a parabola. If the $3$ control points and the quadratic Bézier curve are known, how do you calculate the equation of the parabola (which is an $y=f(x)$ ...
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0answers
11 views

Angle between two lines formed by intersection of two planes with X = 0, Y = 0 and Z = 0 planes

I am new to linear algebra and have been struggling with a solution for this since long. I work on MATLAB. I have a set of points (as in Points3D.mat attached here). I find the equation of best fit ...
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1answer
58 views

Why the existence of automorphism of varieties makes a functor not being a fine moduli space?

Let $F: (Sch) \to (Sets)$ be a functor sends schemes to sets (for example, $F$ sends a scheme $S$ to families of K3 surfaces over $S$ with some fixed polarization). Then it is known that because of ...
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0answers
48 views

Genus of the product of two elliptic curves

In trying to understand the trichotomy of the genus of algebraic curves, I first consider the following two elliptic curves (over $\mathbb{Q}$), well-known to be of rank $2$, $ y^2 = x^3+17$ and $ ...
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0answers
37 views

$\{P\}^-$ notation in Algebraic Geometry

I am reading proposition 2.1.6 in Algebraic Geometry by Robin Hartshorne. Near the end of the paragraph, he defines a function $ \alpha:X \rightarrow t(X) $ where $X$ is a topological space and $t(X)$ ...
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1answer
31 views

Find a projective variety $Z$ and closed subsets $X,Y \subseteq Z$ with $\dim(X)+\dim(Y) \geq \dim(Z)$ and $X \cap Y = \varnothing$

I am trying to find a projective variety $Z$ and closed subsets $X,Y \subseteq Z$ with $\dim(X)+\dim(Y) \geq \dim(Z)$ and $X \cap Y = \varnothing$. However, all my attempts failed. In fact, we ...
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0answers
36 views

Reference to study moduli spaces

I would like to know about references where the problem of finding the infinitesimal deformations of a given geometric structure, and obtaining the corresponding (elliptic?) complex parametrizing the ...
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1answer
76 views

Injectivity Unclear

Let $R=K[x_1,...,x_n]/I$ and $m$ be maximal ideal of $R.$ Let $(s_1,...,s_d)$ be a base of $m/m^2$ where $\dim R_m=\dim_K m/m^2=d.$ Then by Kunz Chapter V.5.10 the canonical epimorphism ...
3
votes
1answer
74 views

Adjoints functors in scheme theory

What's useful information available to us, when we state that : If $ f: X \to Y $ a morphism of schemes, and if $ \mathcal{F} $ denotes $ \mathcal{ O }_X $ - modules, and $ \mathcal { G} $ denotes $ ...
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0answers
33 views

What is a smooth family of divisors?

Suppose that $S$ is a smooth complex projective surface ($\mathbb C$-scheme, reduced, irreducible...). What do algebraic geometers usually mean with the term a smooth family of divisors in $S$? ...
2
votes
1answer
37 views

$j$-invariants of isogenous elliptic curves

Suppose that $E,E'$ are isogenous smooth complex elliptic curves - is there some relation between their $j$-invariants?
2
votes
1answer
35 views

Quotients of curves

Magma (link) has a lot of functionality for computing quotients of curves by group actions. I am interested to know how one does this in general and I am finding it oddly difficult to find literature ...
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1answer
33 views

Subsheaf of a torsion-free sheaf

Let $X$ be a noetherian projective scheme, $\mathcal{F}$ a torsion free $\mathcal{O}_X$-module on $X$ and $\mathcal{G} \subset \mathcal{F}$ submodule. Is it possible that $\mathcal{G}$ is ...
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3answers
45 views

Isomorphism between affine varieties

I am working with a ring and I am trying to show it is not isomorphic (as $k$-algebra) to another ring: $k[x,y,z]/\langle xy-z^2\rangle$ and $k[u,w]$. What I tried so far was. I aim for a ...
2
votes
1answer
60 views

Points of scheme with residue field $k$ vs $k$-point

Let $X$ be a scheme over a field $k$. Consider the following definitions. The residue field of a point $x\in X$ is $k(x)=\mathcal{O}_{X,x}/\mathfrak{m}_x$. The $k$-point of $X$ is the morphism of ...
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0answers
64 views

Why are algebraic cycles rational?

Let $X_{/\mathbb{C}}$ be a projective non-singular variety of dimension $n$ and $Z \subset X$ be an irreductible subvariety of dimension $p$. Denote by $\mathrm{H}_{\mathrm{dR}}^i(X,\mathbb{C})$ the ...
2
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1answer
35 views

Does a simplex with equal altitudes have to be equilateral?

Consider a simplex in $\mathbb{R}^d$. Assume that all its altitudes have the same length. Does it necessarily mean that the simplex is equilateral, i. e. all distances between its vertices are equal ...
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0answers
29 views

Blow up of Hirzebruch surface

I would like to prove that the blow up of $n$-th Hirzebruch surface $F_n=\mathbb P(\mathcal O \oplus \mathcal O(n))$ at a point outside the exceptional section is isomorphic to the blow up of ...
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1answer
114 views

The moduli space of stable maps

I am reading Katz' book Enumerative Geometry and String Theory. I have a few questions regarding the moduli space of degree $d$ genus $0$ stable maps into $\mathbb P^n$, denoted ...
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1answer
39 views

Dual bundle of a stable vector bundle.

Given a vector bundle $E \to X$ over a complex curve $X$, let its rank and degree be $r$ and $d$ respectively. Then the slope of $E$ is defined to be $ \mu(E):= \frac{d}{r}. $ $E$ is defined to be ...
0
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1answer
56 views

Rational normal curve of degree 4

Consider the rational normal curve of degree 4. It can be realized as the intersection of a certain number of quadrics. Usually in examples we consider those quadrics to be singular. However, can at ...
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votes
1answer
83 views

Vector Bundles:differential geometry vs algebraic geometry

I am in trouble about the vector bundle part in the Friedman's book: Algebraic Surfaces and Holomorphic Vector Bundle, I know what is a vector bundle(or a fibre bundle) in the differential geometry, ...