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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9
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1answer
117 views
+100

$p \in C - D$, inflection point for $C$ iff inflection point for $C \cup D$.

Show that if $C$ and $D$ are projective curves in $\mathbb{P}_2$ and $p \in C - D$ then $p$ is a point of inflection for the curve $C$ if and only if $p$ is a point of inflection for the curve $C \cup ...
1
vote
2answers
42 views

Affine open sets of projective space and equations for lines

I am reading Introduction to Algebraic Geometry by Smith et al. and I have some questions about some vocabulary that they use but that is not explicitly defined (I guess it is probably obvious and I ...
1
vote
1answer
35 views

Interpretation of a short exact sequence from elliptic curves in terms of torsors

Consider some elliptic curve $E$ over a number field $k$. Then for any prime $p$ there is a short exact sequence $$ 0 \to E(k)/pE(k) \to H^1(k,E[p]) \to H^1(k,E)[p] \to 0. $$ Now, $H^1$ has an ...
1
vote
0answers
24 views

deducing irreducibility from intersections with hyperplane complements

Let $X$ be a projective variety of $\mathbb{P}^n$. Suppose that for any hyperplane complement $U$ of $\mathbb{P}^n$, $X \cap U$ is irreducible. Then i want to prove that $X$ must be irreducible. Here ...
1
vote
0answers
25 views

Inclusionwise maximal linear subvarieties of a projective variety

Let $X\subseteq\mathbb P^n$ be a complex, projective variety. A linear subspace $L\subseteq\mathbb P^n$ will be called a maximal linear subspace of $X$ if $L\subseteq X$ and for any linear subspace ...
3
votes
2answers
59 views

Deforming line bundles on abelian varieties

Let $X$ ba an abelian variety over $\mathbb C$. I would like to understand how line bundles on $X$ deform. The obstructions to deform line bundles lie in $$\textrm{Ext}^2(L,L)=H^2(X,\mathscr O_X).$$ ...
3
votes
1answer
134 views

The assignment $R\mapsto\operatorname{Iso}_{R\text{-alg}}(A\otimes_k R,M_n(R))$ is a scheme?

Let $A$ be a central simple algebra over some field $k$, with degree $n$. There is a functor $F$ defined by the assignment, for a commutative ring $R$, $$ ...
5
votes
0answers
53 views

Some questions about reduction of elliptic curves

Let $E \rightarrow S$ be an elliptic curve (i.e, a smooth proper curve of genus 1). If $S = \text{Spec (K)}$ where $K$ is a local field, the usual way of doing a reduction at a prime $\mathfrak{p} = ...
3
votes
0answers
49 views

Solving exercise 1.10 in Silverman's AEC

Please note that although there is a very similarly titled question Exercise 1.10 from Silverman "The Arithmetic of Elliptic Curves" this question received no answers. Let $p$ be an odd prime and ...
1
vote
1answer
37 views

Dimension of linear system of divisor of two points on curve of genus greater than 2

This should not be hard, but I am stuck on it nonetheless, so I would much appreciate a solution. Suppose $C$ is a projective non-singular curve of genus $g\geq 2$ and $P,Q$ are distinct points on ...
4
votes
1answer
46 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 ...
1
vote
1answer
30 views

Holomorphic maps between smooth algebraic curves

I am looking for a reference for the following statement: Let $X$ be a smooth projective curve over $\mathbb{C}$. Every holomorphic function $f: X \to \mathbb{P}^1_{\mathbb{C}}$ is in fact a morphism ...
0
votes
0answers
36 views

Affine morphism, Harsthorne ex 5.17 a)

$\newcommand{\Sp}{\text{Spec}}$ Hello, I'm stuck with one question in Hartshorne, exercise 5.17. A morphism between schemes $f: X \to Y$ is affine is there is an open cover $Y = \bigcup V_i$ such ...
0
votes
1answer
25 views

Picard group of Segre Embedding of $\mathbb{P}^1 \times \mathbb{P}^1$

Let $X = V(xy-zw) \subset \mathbb{P}^3$ (the variables are $x,y,z,w$). I know from various sources that $Pic(X) \cong \mathbb{Z} \oplus \mathbb{Z}$, where generators are the two lines $l_x = V(x,w) ...
0
votes
1answer
50 views

Why is the evaluation map of sheaves injective

Let $E$ be a globally generated vector bundle of rank $r$. Let $V$ be a subspace of $H^0(S,E)$ of dimension $r$. We have the evaluation map $ev:V\otimes \mathcal{O}_S\longrightarrow E$. Why is this ...
1
vote
0answers
34 views

What is the significance of Gaitsgory and Lurie's proof of Weil's conjecture for function fields?

Can anyone place this result in context (including historical context) and explain its significance? Is this considered to be a major result?
8
votes
0answers
66 views

Genus of $k(T)$?

Let $F$ be a function field in one variable with total constant field $k$, let $X$ be the set of all places $F$, and let $S$ be a nonempty finite subset of $X$. Then the genus of $F$ is equal to the ...
5
votes
0answers
69 views

algebraically closed fields of characteristic 0 and $\mathbb{C}$

Let $k$ be an algebraically closed field of characteristic 0. Then I've heard that if $k$ has cardinality no greater than that of $\mathbb{C}$, then there is an embedding ...
0
votes
0answers
24 views

j-invariants for an elliptic curve over the Artin ring $k[t]/(t^n)$.

Let $k$ be an algebraically closed field, let $\lambda \in k - \{0,1\}$ and let $C = k[t]/(t^n)$. Hartshorne's "Deformation Theory" chapter 1 exercise 4.9(c) asserts that the family $$y^2 = ...
1
vote
0answers
43 views

Showing an Ideal is Irreducible (alternative proof)

Recently, this question was posted regarding the following: Question: Show $$(x^3,y^5,z^2)\subset\mathbb{C}[x,y,z]$$ is an irreducible ideal. I was wondering if the following could be reviewed ...
2
votes
1answer
32 views

Why are equivariant morphisms of $G$-torsors necessarily isomorphisms?

This was something I read on the Stacks project, but whose proof was omitted. Simply stated, if $f\colon E\to F$ is a $G$-equivariant morphism of $G$-torsors over a scheme $X$, why is $f$ ...
1
vote
1answer
44 views

Proposition 2.24 on Liu Qing‘s AG book

One of the conditions of this prop is $f:Y\longrightarrow X$ is a closed immersion. This makes $f(Y)$ is closed in $X$. Then we have $ (f_*\mathcal{O}_Y)_x = \begin{cases} 0, & \mbox{if }x ...
2
votes
0answers
48 views

Dimensions of quotient rings of $K[x,y]$

I have tried to solve the following problem and would be very grateful if someone could check my answer. Let $K$ be an algebraically closed field with $\mathrm{char}(K)=0$. I wish to compute ...
2
votes
1answer
39 views

Is every codimension one subvariety of a projective variety a set-theoretic complete intersection?

Let $X$ be a projective variety over $\mathbb C$ and $D\subseteq X$ some subvariety which is pure of codimension one. In fact, in my case $D$ is the complement of an open affine subvariety $U\subseteq ...
2
votes
1answer
51 views

$R^nf_*\mathbb{Z}$ trivial for a morphism with hypersurface fibers.

I have some questions on local systems. If $f:X\to Y$ is a morphism of projective complex algebraic varieties, $Y$ being a curve, I want to prove that if the fibers of $f$ are smooth hypersurfaces in ...
4
votes
1answer
68 views

Smooth points with obstructed deformations

Let $k$ be an algebraically closed field, e.g. $k=\mathbb C$. Let $Art_k$ be the category of local Artin $k$-algebras with residue field $k$. A deformation functor is a functor $D:Art_k\to Sets$ such ...
2
votes
1answer
14 views

Strongly convex cones

A polyhedral cone is strongly convex if $\sigma \cap -\sigma =\{0\}$is a face. Then here is the following proposition. Let $\sigma$ be a strongly convex polyhedral cone. Then the following are ...
3
votes
2answers
43 views

Scheme whose points over $x\colon\mathrm{spec}(R)\to X$ are the isomorphisms $x^*(F)$ and $x^*(E)$?

If one has two vector bundles $E\to X$ and $F\to X$ over a scheme $X$, why is there a scheme $S$ over $X$ with points of $S$ over a point $x\colon\mathrm{spec}(R)\to X$ is precisely the set of ...
4
votes
1answer
52 views

Varieties and ideals

I'm doing the exercises from Fulton of Algebraic Geometry and I'm stuck in the problem 2.44 Let $V$ be a variety in $\mathbb{A}^{n}$, $I=I(V)\subset k[x_{1},\ldots,x_{n}]$, $P\in V$ and let $J$ be ...
3
votes
0answers
55 views

Motive of a curve and its Jacobian

Let $C$ be a smooth projective curve with a $k-$rational point $x_0$ and $J$ its Jacobian variety. Let us consider the (almost) canonical embedding $j:C \to J$ that sends $x_0$ to the identity $e \in ...
2
votes
0answers
28 views

etale morphism between sheaves

We knoe that if $f$ and $ f\circ g$ are both etale morphisms between schemes, then so is $g$. Does this statement hold for etale morphisms between sheavs on etale site over a scheme? More generally, ...
0
votes
0answers
50 views

Any quartic in $\mathbb P^3$ contains only finitely many lines.

I want to prove thath any quartic $X$ in $\mathbb P^3$ contains finitely many lines, but I don't know any method for computing lines on a surface. What is the idea of the proof?
1
vote
0answers
17 views

Amoeba of a line in the plane: An example

Let $z+w+1=0$ a line in $\mathbb{C}^2$ and let $x=log|z| \ge 0$ and $y=log|w|$. I have to show that $$ log(e^x-1) \le y \le 1+e^x $$ But I can't do it! Can you help me, please?
0
votes
1answer
11 views

Is there an explicit construction of an embedding of Severi Brauer varieties $B(A)\times_k B(A')\hookrightarrow B(A\otimes_k A')$?

This question arose from a note on Galois descent. If $A$ is a central simple algebra over a field $k$, let $B(A)$ be its Severi Brauer variety. There is always at least a morphism $$B(A)\times_k ...
3
votes
0answers
39 views

Normal projective varieties and its coordinate ring

Let $k[X_0,...,X_n]$ be a polynomial ring over an algebraically closed field of characteristic zero and $I$ an ideal of $k[X_0,...,X_n]$ generated by homogenous polynomials. Denote by $X$ the ...
3
votes
1answer
47 views

Introductory book on Gromov Witten Theory

I am looking for a good introduction to Gromov Witten Theory. I have a background in algebraic geometry, which corresponds roughly to the first three chapters of Hartshorne. Thanks in advance!
3
votes
1answer
44 views

Question about certain morphism between affine spaces

Let the map $\varphi:\mathbb{A}^2\to\mathbb{A}^4$ is given by $$(x,y)\mapsto(x, xy, y(y-1), y^2(y-1)).$$ How to find the system of equations, which defines the image of $\varphi$? If we denote ...
1
vote
1answer
44 views

Exercise about Zariski-topology

I'm trying to became familiar with the basic notions of algebraic geometry and I proved the fact which states every continuous mapping from $\mathbb R^n$ with the Zariski topology to $\mathbb R$ with ...
0
votes
0answers
29 views

Cohomology of conic bundle 3-folds

It is known that for a smooth cubic 3fold $X\subset \mathbb{P}^4$ we have $H^3(X,\mathcal{O}_X)$ (or if you prefer $H^{0,3}(X)=0$). Moreover, if I project off a line $l\subset X$ I can resolve the map ...
0
votes
0answers
51 views

Serre's criterion and closure

Let $X$ be a projective scheme of pure dimension $n \ge 2$. Let $U$ be an open dense subset of $X$ such that $\mathrm{codim}(X\backslash U,X) \ge 2$ and for all points $x \in U$, $\mathcal{O}_{X,x}$ ...
1
vote
0answers
43 views

Is exceptional divisor always a Projective bundle over the centre?

Let $f:\tilde X \rightarrow X$ be a blow up at center Z. Is $f^{-1}(z)=\mathbb{P}^{k}$ ?for some $k$, $\forall$ $z\in Z$ In the case of blow-up of $\mathbb{A}^{n}$ at origin it is very clear that the ...
3
votes
0answers
48 views

Regarding the connected component of $|1/J| < 1$ containing $\infty$

How does one explicitly describe the connected component of $|1/J| < 1$ containing $\infty$? Here, $J = J(\tau) = j(\tau)/12^3$ is the normalized $j$-invariant so that $J(i) = 1$, and $\tau$ is in ...
0
votes
1answer
31 views

Question related to dimension of a manifold of zeros over $\mathbb{R}$

Let $F \in \mathbb{z}[x_1,\ldots, x_n]$ be a form of degree $d>1$. Let $V_{\mathbb{R}}$ be the manifold of real zeros of $F$ (I am using the notation of an article I am reading here). Let $B = ...
1
vote
1answer
51 views

Ideal sheaf of intersection of two surfaces in $\mathbb P^3$

Let X be an intersection of 2 surfaces of degree $d_1,d_2$ in $\mathbb P^3$. Is it true that there is a short exact sequence $$ ...
-1
votes
0answers
39 views

Easy ways to calculate $\dim \mathbb{K}[x,y]/(f,g)$ [closed]

Let $f,g \in \mathbb{K}[x,y]$ be to polynomials without common divisors and $\deg f = n, \deg g = m$. I want to prove that $\dim\mathbb{K}[x,y]/(f,g) \leq n\cdot m$. I know only that $|V((f,g))| \leq ...
0
votes
0answers
91 views

Notation on De Jong and Starr's paper

I reading De Jong and Starr's paper "Divisor classes and the virtual canonical bundle for genus 0 maps". I am quite confused about what would be the correct definition of functor ...
1
vote
0answers
70 views

A smooth rational curve of degree 4 in $\mathbb{P}^3$ is contained in a unique smooth quadric surface

I am trying to solve the exercise IV.6.1 from Hartshorne's "Algebraic geometry": A smooth rational curve of degree 4 in $\mathbb{P}^3$ is contained in a unique quadric surface $Q$, and $Q$ is ...
0
votes
0answers
14 views

Computing local coordinates

Let $p=[x_0,y_0,1]\in \mathbb P_2(k)$ (projective space). Determine a projective transformation $\phi\in GL(3,k)$ such that $\phi(p)=[0,0,1]$ and name the coordinates explicit. Its easy to see that ...
1
vote
1answer
30 views

For $0\to F\to E\stackrel{\varphi}{\to} L\to 0$, why is the pullback under $\varphi$ of a constant section an $F$-torsor?

I think the following is used in classifying $F$-torsors. Let's take $0\to F\to E\stackrel{\varphi}{\to} L\to 0$ to be an exact sequence of vector bundles, all over a scheme $S$, where ...
2
votes
1answer
60 views

Help with Math software (macaulay 2)

I just started working with Macaulay 2 and need some help. I need to find the number of solutions of a system of equations. I am having difficulty imputing this into the software so please be specific ...