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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5
votes
2answers
149 views

Showing an analytic map has closed irreducible image

Let $X,Y$ be complex algebraic varieties with $X$ (algebraically hence also analytically irreducible), $\pi : Y \to X$ an algebraic map with each fiber a finite set, and $g:X \to Y$ an analytic map ...
3
votes
1answer
92 views

Union of Hyperplanes

Suppose $f=f(x)=f(x_1,\cdots,x_n)$ is a homogeneous polynomial of degree $n$ in the ring $k[x_1,\cdots,x_n]$, with $k$ a field. Denote by $H_f$ the hypersurface given by $f=0$. Suppose that for every ...
6
votes
1answer
119 views

Fibers equal implies schemes equal in a neighborhood

Let $f:X \rightarrow Y $ be a morphism of locally Noetherian schemes. Let $Z$ be a closed subscheme of $X$ and suppose that there exists a point $y \in Y$ such that $Z_y=X_y$ as schemes. Show that if ...
4
votes
3answers
155 views

Does the inclusion from affine schemes into schemes preserve pushouts?

Let $K$ be a field. What is an example of two $K$-algebra morphisms $R\to T$ and $S\to T$ such that $\operatorname{Spec}(R\times_T S)$ is not the pushout of the diagram $$ ...
3
votes
2answers
134 views

“The notion of an affine algebraic set is still not satisfactory”

In their book "Algebraic Geometry I" Gortz and Wedhorn at page 16, after defining the category of affine algebraic sets (Zariski topology in $\mathbb A^n_k$, corrispondence between radical ideals and ...
4
votes
2answers
106 views

How can we check the gluing property of sheaf of ideals?

For a ringed space $(X,\mathcal{O}_X)$, one can define a sheaf of ideals $\mathcal{J}$ of $\mathcal{O}_X$. Then how can we see the $\mathcal{J}$ satisfies the conditions of sheaf? Especially, I cannot ...
3
votes
2answers
110 views

Question about proof of Hartshorne book Lemma III.2.4

Let $(X,\mathcal{O}_X)$ be a ringed space. For any open subet $U \subseteq X$, Let $\mathcal{O}_U$ denote the sheaf $j_!(\mathcal{O}_X|_U)$ which is the restriction of $\mathcal{O}_X$ to $U$, extended ...
7
votes
2answers
221 views

Nullstellensatz and the Fundamental Theorem of Algebra

I came across an interesting problem that basically said something along the lines of ``Show that Hilbert's Nullstellensatz is equivalent to the Fundamental Theorem of Algebra.'' My algebraic geometry ...
7
votes
2answers
360 views

Derived Category of Coherent Sheaves on Elliptic Curves

I know little about algebraic geometry, however while studying noncommutative geometry some results showed that a category I understand well (holomorphic vector bundles over noncommutative tori) was ...
0
votes
0answers
39 views

How can I reformulate my problem to make it convex?

I would like to find the symmetric positive definite matrix $S\in \mathcal{M}_{m,m}$ that minimizes the function $f(S)=\mathrm{trace}(S)+m^2\mathrm{trace}(S^{-2})$ which has been proven to be convex ...
2
votes
1answer
41 views

Example s.t. $\phi^* \circ\mathbb{V}(\mathfrak{b})\subsetneq \mathbb{V} (\mathfrak{b}^c)$

Let $\phi:A\to B$ be a $\mathtt{CRng}$ morphism. It can be shown that $\overline{\phi^* \circ \mathbb{V}(\mathfrak{b})}=\mathbb{V}(\mathfrak{b}^c)$ where $\phi^*: \mathtt{Spec}(B)\to ...
-1
votes
1answer
169 views

Algebraic Curves

Let $F$ be a non-constant polynomial in $k[X_1,...,X_n]$, $k$ algebraically closed. Show that $\mathbb A^n \setminus \mathrm{V}(F)$ is infinite if $n\geq 1$, and $\mathrm{V}(F)$ in infinite if ...
5
votes
1answer
200 views

Translation of a Paper of Tate

I'm just wondering if anyone knows if the paper "Classes d'isogenie des varietes abeliennes sur un corps fini" by John Tate has been translated into English. My French is not that good and I found it ...
2
votes
1answer
80 views

Local rings and classifying singularities

My query is a little vague, but I'll try to be as concrete as possible. Is there some sense in which the local ring of an algebraic variety (or more general complex space) at a point depends only on ...
3
votes
1answer
120 views

Scheme glued out of spectra of local rings

This is a follow up question to this question. Is every scheme over a field $K$ the colimit (over some arbitrary complicated diagram) of affine schemes $\operatorname{Spec}(R_\alpha)$ where each ...
3
votes
1answer
51 views

Does this diagram of Chern classes and push forwards commute

Let $p:Y\to X$ be a birational proper surjective morphism of regular surfaces, and let $D$ be a divisor on $Y$ such that $p(D)$ is a point. Then $p_\ast D =0$ by definition. Is there an easy way to ...
4
votes
2answers
99 views

Where do I use the fact that $F$ is algebraically closed in this proof?

I have to do the following. Let $F$ be an algebraically closed field. $I\in F[X_1,...,X_n]$ an ideal. Denote by $S(I)$ the subset in $F^n$ consisting of all $n$-tuples $(a_1,...,a_n)\in F^n$ such that ...
3
votes
1answer
92 views

Formula for index of $\Gamma_1(n)$ in SL$_2(\mathbf Z)$

Is there a precise formula for the index of the congruence group $\Gamma_1(n)$ in SL$_2(\mathbf Z)$? I couldn't find it in Diamond and Shurman, and neither could I find an explicit formula with a ...
7
votes
2answers
206 views

Adjunction for varieties with higher codimension

For $X \subseteq \mathbb{P}^n$ a smooth hypersurface, the canonical divisor $K_X$ can be computed as $$ K_X = (K_{\mathbb{P}^n} + X)|_X. $$ Is there a similar formula where $X$ is of higher ...
3
votes
0answers
77 views

What is the Induced Representation in Geometric Terms

As is well known, for $G$ a Lie group, and $H$ a subgroup of $G$ such that $G/H$ is homogeneous space (or maybe this is always a homogeneous space?), we have a correspondence between representations ...
11
votes
1answer
320 views

Torsion Chern class?

Can somebody give an example of a complex manifold whose first Chern class is a torsion class? In general it seems that Chern classes may have torsion part as well as free part. However when using ...
2
votes
1answer
199 views

Spec R is irreducible

A topological space is called reducible if $X=X_1\cup X_2$ for two closed subsets $X_1,X_2$ with $X_1\ne X\ne X_2$. Otherwise its called irreducible, want to show that $\text{Spec}(R)$ is irreducible ...
7
votes
1answer
216 views

Singularities of Curves in Positive Characteristic

Given a collection of polynomials $\mathscr{F}\subset\mathbb{Z}[x_1,\ldots,x_n]$, we can associate to each prime ideal of $\mathbb{Z}$ an affine variety as follows: $$ (p)\longmapsto ...
2
votes
1answer
215 views

Fiber of morphism of integral schemes

Let $X$ and $Y$ be integral schemes and $f : X \to Y$ a morphism. Let $X_{\eta'}$ be the fiber of $f$ over the generic point $\eta'$ of $Y$, i.e. the base change $X_{(K)} = X \times_Y K$ where $K = ...
5
votes
1answer
197 views

Must two morphisms into an algebraic space which agree on closed points be the same?

First I apologize if this is elementary. I have just started looking at the basics of stacks and algebraic spaces so my understanding is lacking. Let's work over an algebraically closed field $k$. ...
9
votes
3answers
698 views

A cohomological statement equivalent to the Riemann Hypothesis

Is there a possibility for looking for a theory of cohomology and an equivalent cohomological statement for Riemann hypothesis over $\mathbb{Z}$?
5
votes
1answer
85 views

hypersurface intersected with generic line

Let $f \in \mathbb{R}[x_1,\cdots,x_m]$ be a homogeneous multivariate polynomial of degree $n$. Now, for $u,v \in \mathbb{R}^n$ the form $f(\lambda u + \mu v)$ can be written as $\mu^n h(\lambda/\mu)$, ...
2
votes
1answer
170 views

Arithmetic genera of thickened curves

Let $X$ be a smooth projective curve of genus $g>0$ and $L \to X$ a line bundle of degree $d>2g-1.$ Let $\mathcal{I}_X$ be the ideal sheaf of $X \hookrightarrow L$ (embedded by the zero ...
5
votes
2answers
89 views

some question of $\mathcal{O}_X$-module

Let $X$ and $Y$ be schemes and let $F$ be a sheaf on $Y$. Let $f: X \rightarrow Y$ be a morphisme of schemes. Define the inverse image of $F$, $$f^*F:= ...
2
votes
1answer
165 views

Composition of dominant rational maps is dominant

Let $f:V\dashrightarrow W$ and $g:W\dashrightarrow X$ be two rational dominant maps of affine varieties, where dominant means $f(\mathrm{dom} \, f)$ dense in $W$ ang $g(\mathrm{dom} \, g)$ dense in ...
1
vote
1answer
93 views

Singular points

For an algebraic variety $Y\subset\mathbb{C}^d$ defined by $$ Y=\{z\in\mathbb{C}^d : \ f_1(z)=\dots=f_r(z)=0\} $$ we say that $y\in Y$ is a singular point if the rank of the complex Jacobian of ...
4
votes
1answer
234 views

Join and Zariski closed sets

A set in $\mathbb{C}^n$ is called Zariski-closed if it can be written as the set of zeroes of some set of polynomial equations $$ V(f_1,...,f_m) = \left\{ z \in \mathbb{C}^n \mid f_1(z)=...=f_m(z)=0 ...
5
votes
1answer
96 views

A question on generic point and a question on Hartshorne

On page 134, Weil divisors, example 6.5.2, he said: "The divisor of $y$ is $2Y$, because $y=0$ implies $z^2=0$, and $z$ generate the maximal ideal of the local ring at the generic point of $Y$." I was ...
2
votes
1answer
70 views

Product of two regular varieties over an imperfect field

I am trying to find a counterexample to the following, but am unable to find one. Any help would be appreciated, and also an explanation of why it works. I am trying to show that over an imperfect ...
9
votes
1answer
232 views

Ext between two coherent sheaves

Let $X$ be a smooth projective variety over a field $k = \overline k$. From Hartshorne we know, that $\textrm{dim} \, H^i (X,F)<\infty$ for any coherent sheaf $F$. How to show, that all $Ext^i ...
2
votes
1answer
243 views

Problem with drawing ellipse with code.

I am trying to draw an ellipse. This is the code. The logic I am using is to start with "Bigger" radius, and then uniformly decreasing it to "Smaller" radius. But i am getting this output : that ...
12
votes
2answers
1k views

Good problems in Algebraic Geometry

I am now using Fulton's book Algebraic Curves to learn algebraic geometry from and have just finished chapter 2. However I feel that the problems are not very inspiring (at the moment at least) and ...
4
votes
1answer
154 views

How to show $\phi(X)$ contains a non-empty open set of its closure $\overline{\phi(X)}$?

From T.A.Springer, Linear Algebraic Groups, the end of Chapter 1. Assume $X\rightarrow Y$ is a morphism of varieties. Using a covering of $Y$ by affine open sets, we reduce the proof to the case ...
4
votes
0answers
90 views

Application of Hirzebruch Riemann Roch

Let $X,Y$ be smooth projective varieties over $\mathbb C$ where $X$ is the universal cover of $Y$. Assume that the fundamental group of $Y$ is finite and has order $d$. Then we want to show that $\chi ...
5
votes
1answer
158 views

Betti Numbers and the Néron-Severi group.

In the Paper "On the Mordell-Weil lattices" it is proved that the rank $\rho$ of the Néron-Severi group of a rational elliptic surface equals 10. Without any further explanation, it is stated that ...
3
votes
0answers
85 views

Decomposing Semisimple Perverse Sheaves

Assume $\mathbf{G}$ is an algebraic group over an algebraic closure $\overline{\mathbb{F}_p}$ for some prime $p>0$. Let $\mathscr{M}\mathbf{G}$ be the category of all ...
0
votes
1answer
87 views

Degree of sum algebraic functions

This question I have asked on mathoverflow already: http://mathoverflow.net/questions/123921/degree-of-sum-algebraic-functions Let $C$ - curve, $f_1, f_2 \in K(C)$. How to prove that deg$(f_1 + f_2) ...
6
votes
1answer
140 views

If a group scheme $G$ operates on another scheme $X$, how do you define orbits?

In my specific case, $G=\mathrm{Spec}(k[M])$ is an algebraic torus acting on a toric variety $X_\Sigma$ corresponding to a fan $\Sigma$ when $k$ is not necessarily algebraically closed (or maybe even ...
5
votes
3answers
294 views

Is $(xy-1)$ a maximal ideal in $\mathbb C[x,y]$

I learnd that the maximal ideals in $\mathbb C[x,y]$ have the form $(x-z_1, y-z_2)$ by the Nullstellensatz. But if we set $I=(xy-1)$ then $\mathbb C[x,y]/I$ is isomorphic to $\mathbb C[x,1/x]$ which ...
0
votes
0answers
50 views

Do “multiples” of open dense sets of an algebraic group union to the whole group?

Let $G$ be an algebraic group. From algebraic geometry we know there exists an open dense subset $U$ of $G$ such that $U$ is nonsingular. Since left multiplication of $U$ by elements of $G$ is an ...
5
votes
1answer
140 views

Finite bijective morphism to variety with separable function field

Let $V$ be an $n$-dimensional variety over $k$. The function field $k(V)$ doesn't have to be separable over $k$ but I'd like to know which conditions imply that we can find a finite bijective morphism ...
6
votes
1answer
276 views

How much do I need to learn before I can read about Toric varieties?

I have a copy of the book "Introduction to Toric varieties" by William Fulton, and over the next few months I'd like to make some progress on it. As a first goal, I'd like to be able to read just ...
2
votes
1answer
128 views

Over $\mathbb{R}$, if $Z(p') \subset Z(p)$ when does $p' \vert p$?

I'm mainly wondering about the planar case, when $p', p \in \mathbb{R}[x,y]$. For instance the simplest case would be when $Z(p')$ is a line contained in $Z(p)$, does it follow that $p' \vert p$? I ...
5
votes
0answers
160 views

Coarse moduli space with no autmorphisms is also a fine moduli space

I'm working in the category of schemes over an algebraically closed field $k$, $Sch_k$. Suppose I have a contravariant functor $F:Sch_k\rightarrow (Set)$ which has a coarse moduli space $M$ (which is ...
6
votes
1answer
103 views

A new(?) partial order on the set of continuous maps

Let $X,Y$ be topological spaces. Define a partial order on $\hom(Y,X)$ as follows: $f \leq g$ if $f^{-1}(U) \subseteq g^{-1}(U)$ for all open subsets $U \subseteq X$. Equivalently, $f(y)$ is a ...