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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4
votes
1answer
146 views

If the special fiber of a flat morphism is reduced, then any other fiber is reduced?

Suppose $R=\mathbb{C}[x_1,\ldots, x_n]$ is a polynomial ring with $I$ being an ideal of $R$. Let $I'$ be an ideal of $R[t]$. If $R[t]/I'$ is flat as a $\mathbb{C}[t]$-module and over $0$, ...
5
votes
1answer
268 views

trivial Picard group

let $S=\operatorname{Spec}(A)$ be an affine scheme. For which ring $A$, not field is it known that $H^1(S,\mathcal{O}_S^{*})$ is trivial? If $X\to S$ is a finite map and $H^1(S,\mathcal{O}_S^{*})$ is ...
9
votes
3answers
893 views

Krull dimension of $\mathbb{C}[x_1, x_2, x_3, x_4]/\left< x_1x_3-x_2^2,x_2 x_4-x_3^2,x_1x_4-x_2 x_3\right>$

Krull dimension of a ring $R$ is the supremum of the number of strict inclusions in a chain of prime ideals. Question 1. Considering $R = \mathbb{C}[x_1, x_2, x_3, x_4]/\left< x_1x_3-x_2^2,x_2 ...
2
votes
1answer
190 views

Congruence subgroups and modular curves of type (M,N)

I would like to study the "modular curve" $Y(M,N)$, parametrizing an elliptic curve $E$ together with $p \in E[M]$ and $q \in E[N]$ (here and in the following $M$ divides $N$). Let $\Gamma(M,N)$ be ...
1
vote
1answer
388 views

relating flatness, equidimensional, and complete intersection

I am a bit confused and am trying to clarify some notions. First consider the following well-known statement. A dominant map $f:X\rightarrow Y$ between regular varieties is flat if and only if it is ...
2
votes
1answer
80 views

Extension of morphisms on surfaces

Consider two regular integral proper algebraic surfaces $X$ and $Y$ over a DVR $\mathcal O_K$ with residue field $k$. Let $U \subset X$ be an open subset, s.t. $X\setminus U$ consists of finitely many ...
1
vote
0answers
71 views

Partial fraction expansion of rational functions

I've been trying to read about partial fraction expansion of rational function. Is the following statement equivalent to the uniqueness+existence of partial function expansion?: Let $\mathbb{F}$ be ...
4
votes
0answers
81 views

the existence of a closed subset of $\mathbb{A}^8\times \mathbb{A}^1$ which is flat over $\mathbb{A}^1$

I've been doing some reading on deformation theory and one way it is used is to study singularities on varieties while perturbing the varieties. I would actually like to use deformation theory to ...
2
votes
1answer
155 views

Divisor of degree 2 on a smooth plane curve

Let $X$ be a smooth plane curve of genus $3$ (assume a smooth plane quartic) and $D$ a divisor of degree $2$ on this curve. Assume that $\mathcal{l}(D)>0$. It means that there exists a rational ...
3
votes
1answer
109 views

How to handle group schemes by points?

I find it is very inconvenient to handle group schemes by its defination(i.e. everything is defined by morphism). And I have noticed that for group varieties, one can treat them as actual groups(i.e. ...
0
votes
1answer
49 views

Reducing a flat morphism $\psi:X\rightarrow\mathbb{A}_{\mathbb{C}}^1\;$ to $\;\psi|_{Y}: X\cap Y\rightarrow \mathbb{A}_{\mathbb{C}}^1$

Suppose $\psi: X\rightarrow \mathbb{A}_{\mathbb{C}}^1$ is a flat morphism, where $X\subseteq \mathbb{A}_{\mathbb{C}}^n$ with $X$ not needing to be smooth, with $\psi^{-1}(0)$ being a complete ...
0
votes
1answer
77 views

explicitly constructing a certain flat family

Is it possible to construct a flat family $$ \phi:\mathbb{A}_{\mathbb{C}}^8=\operatorname{Spec} \mathbb{C}[x,y,z,w,a,b,c,d]\longrightarrow \operatorname{Spec} \mathbb{C}[t_1, t_2, t_3] ...
0
votes
0answers
1k views

Converting standard equation for a paraboloid to a parametric one

I have the equation for a hyperbolic paraboloid in $x$, $y$, and $z$: $$\frac{z}{c} = \frac{x^2}{a^2} + \frac{y^2}{b^2}$$ I also have the parametric equations for the same parabaloid: $$x = a u ...
2
votes
0answers
59 views

Descent through blow-up

Let $X$ be a variety with $Y \subsetneq X$ a proper closed subvariety. Let $Z$ denote the blow-up of $X$ along $Y$. Let $f: Z \rightarrow X$ be the canonical map. Suppose that we have a coherent sheaf ...
2
votes
1answer
82 views

If $\phi^{-1}(0)$ in $\phi:X\rightarrow Spec\; \mathbb{C}[t]$ is a complete intersection, then is $\phi$ flat?

This is a simple question so I am hoping the answer is quite simple as well. Suppose $\phi:X\rightarrow Spec\; \mathbb{C}[t]$ is a map such that the algebraic variety $\phi^{-1}(0)$ is a complete ...
4
votes
2answers
157 views

zeroes of forms on Riemann surfaces

Let $P$ be a point on a Riemann surface. Does there exist a non-trivial differential form $\omega$ on $X$ such that $\omega$ vanishes at $P$? Does there exist a non-constant rational function $f$ on ...
1
vote
2answers
108 views

Regular functions on $\mathbb P_k^n$

Let be $k$ an algebraically closed field and let's consider a projective algebraic set $V\subseteq\mathbb P^n_k$ with the induced Zariski topology. If $U\subseteq V$ is open, likewise the affine ...
3
votes
0answers
223 views

There is some intuitive idea of Pascal's 's theorem in Projective Geometry?

In projective geometry, Pascal's theorem (formulated by Blaise Pascal when he was 16 years old) determines that a hexagon inscribed in a conic, the lines that contain the opposite sides intersect in ...
6
votes
2answers
195 views

Actually calculating an Intersection of Variety and Divisor

This has been bugging me for a while now. Say I have a projective variety given by some polynomial $P$ and the canonical divisor of the projective space. How can I concretly calculate the ...
0
votes
1answer
137 views

References for Vector bundle over a projective space?

I know just the basics of sheaf theory and would like to ask about good references for "vector bundle over a projective spaces"?
3
votes
1answer
71 views

Separatedness of a composition, where one morphism is surjective and universally closed.

I'm stuck with the following problem: Let $f:X \rightarrow Y$ and $g:Y \rightarrow Z$ be scheme morphisms such that f is surjective and universally closed and such that $g \circ f$ is separated. The ...
1
vote
2answers
111 views

quotients of curves by actions of roots of unity

let $X$ be a smooth projective irreducible curve of genus $g$ over the complex numbers. Assume that $X$ comes with an action of $\mu_d$. Is the quotient $Y:=X/\mu_d$ always smooth? Let $\pi: X \to ...
1
vote
1answer
237 views

Sheaf of rings with vanishing stalk?

How common is that a sheaf of rings has a vanishing stalk? To define the rank of a locally free sheaf of $\mathscr{O}$-modules, for instance, $\mathscr{O}_x=0$ may cause some problem, since the rank ...
5
votes
1answer
236 views

Taking stalk of a product of sheaves

Let $(\mathscr{F}_\alpha)_\alpha$ be a family of sheaves on $X$, and $\prod_\alpha\mathscr{F}_\alpha$ the product sheaf. If $x\in X$, is it true that ...
3
votes
1answer
303 views

Is locally free sheaf of finite rank coherent?

Let $\mathcal{F}$ be a locally free sheaf of finite rank of scheme $X$, is $\mathcal{F}$ coherent? By the definition of locally free sheaf, there exists an open cover {$U_i$} of $X$ such that ...
4
votes
1answer
146 views

Cech cohomology of $\mathbb A^2_k\setminus\{0\}$

I'm trying to prove, via the Cech cohomology, that $S=\mathbb A^2_k\setminus\{0\}$ with the induced Zariski topology is not an affine variety. Consider the structure sheaf $\mathcal O_{\mathbb ...
4
votes
1answer
220 views

codimension of “jumping” of the dimension of fibers

Let $f:X\rightarrow Y$ be a dominant morphism of projective (and smooth if you like) varieties over an algebraically closed field $k$ such that $n=\dim(X)=\dim(Y)$. Then $f$ is proper, so by ...
2
votes
0answers
61 views

Restriction of locally free sheaf associated projective modules

My question comes from the paper of Tamafumi's "On Equivariant Vector Bundles On An Almost Homogeneous Variety" (it can be downloaded freely in ...
3
votes
1answer
174 views

Exact sequence in Beauville's “Complex Algebraic Surfaces”

On page 3 of Beauville's book (Lemma I.5) he takes two curves $C$ and $C'$ in a surface $S$ an takes global sections $s\in H^0(S,\mathcal{O}_S(C))$ and $s'\in H^0(S,\mathcal{O}_S(C'))$. In a recent ...
4
votes
0answers
139 views

Varieties and Statistics

Consider a random variable $X$ that can take on the values $0,1$ and $2$. So we have $$p_i = P(X=i), \ i = 0,1,2$$ $$\sum_{i=0}^{2} p_i = 1$$ and $$0 \leq p_i \leq 1$$ So identifying a random variable ...
7
votes
1answer
224 views

Exact sequence of sheaves in Beauville's “Complex Algebraic Surfaces”

On the first pages of Beauville's "Complex Algebraic Surfaces", he has a surface $S$ (smooth, projective) and two curves $C$ and $C'$ in $S$. He defines $\mathcal{O}_S(C)$ as the invertible sheaf ...
8
votes
1answer
133 views

Line Bundle on subvarieties

I've been having problem actually restricting a Line bundle $L$ defined on some projective space $\mathbb C \mathbb P^{N-1}$ to a subvariety $X$. I know how to do this on an abstract level, but ...
1
vote
0answers
36 views

trivial actions

Let $X$ be a smooth, projective, connected algebraic variety defined over a subfield of $\mathbb{C}$. Assume $X$ is equipped with an automorphism $g: X \to X$. By functoriality we get morphisms ...
1
vote
1answer
116 views

Solving weighted power sum equations

I am thinking about this problem: Let $\sum_{i=1}^N a_ix_i^k = b_k,\ k = 0,1,2,\dots,$ be an equation system. If I only know all the values of $b_k$'s, is there any way to find out the values of ...
3
votes
1answer
97 views

Relating the normalization of a variety over $\mathbb{Z}$ to its reduction mod $p$.

I'm looking for a proof (or better, a reference) for the following claim. Claim: Let $X$ be an (irreducible) variety defined over $\mathbb{Z}$. Let $\nu:\tilde{X}\rightarrow X$ be its normalization. ...
4
votes
0answers
64 views

local system attached to a finite morphism

Imagine you have a finite proper morphism between smooth projective complex curves, say $f: X \to Y$. Denote by $S$ the image of ramification points and by $d$ the degree of $f$. Then $f_\ast ...
12
votes
1answer
465 views

Regular local ring and a prime ideal generated by a regular sequence up to radical

Let $R$ be a regular local ring of dimension $n$ and let $P$ be a height $i$ prime ideal of $R$, where $1< i\leq n-1$. Can we find elements $x_1,\dots,x_i$ such that $P$ is the only minimal prime ...
4
votes
1answer
99 views

If a principal divisor is defined over K, then is the function?

Let $X$ be an algebraic variety, $D$ a principal divisor of $X$ defined over $K$, i.e. the points of $D$ are in $X(K)$ and there is a function in $\overline{K}(X)$ whose divisor is $D$. Is $D$ ...
2
votes
0answers
154 views

symmetries of families of polynomial functions

The family of quadratic functions $F_2(a,b,c)$, consisting of all functions of the form $f(x)=ax^2+bx+c$, has the nice property (call it P) that given any $f,g\in F_2$, there is a sequence of function ...
13
votes
1answer
1k views

What is an intuitive meaning of genus?

I read from the Finnish version of the book "Fermat's last theorem, Unlocking the Secret of an Ancient Mathematical Problem", written by Amir D. Aczel, that genus describes how many handles there are ...
3
votes
1answer
101 views

Homogenous polynomials

In section 3.1 (3rd paragraph on page 4) in this paper, I cannot understand why $Q$ and $R$ are homogeneous: (Given $A$, $B$ are homogenous. Capital letters denote homogenous polynmials.)
2
votes
1answer
109 views

Restricting Line Bundles to Hypersurfaces?

The Adjunction Formula is given to be $K_V = (K_X \otimes [V])_V$ Where $K_V$ is the canonical class on $V\subset X$ and $K_X$ of $X$. And $[V]$ denotes the line bundle associated to $V$. Now say, ...
3
votes
3answers
142 views

Exposition on hyperelliptic curves

Is there any literature that introduces hyperelliptic curves without the view towards cryptography? Even better is if there are any books that talk about them with (about) the same amount of detail as ...
4
votes
1answer
220 views

Direct limit of localizations of a ring at elements not in a prime ideal

For a prime ideal $P$ of a commutative ring $A$, consider the direct limit of the family of localizations $A_f$ indexed by the set $A \setminus P$ with partial order $\le$ such that $f \le g$ iff ...
1
vote
1answer
104 views

smooth quotient

Let $X$ be a smooth projective curve over the field of complex numbers. Assume it comes with an action of $\mu_3$. Could someone explain to me why is the quotient $X/\mu_3$ a smooth curve of genus ...
3
votes
1answer
186 views

Restriction of flat morphism

Suppose that $f\colon X\to Y$ is a flat morphism of varieties over an algebraically closed field $k$. Let $E\subseteq X$ and $F\subseteq Y$ be closed subvarieties such that $f(E) = F$. Is it true that ...
3
votes
1answer
106 views

Product of varieties

If we have two rational varieties (i.e varieties which are birational to some projective space) is their product also a rational variety? would this rely on the fact that the Zariski topology is finer ...
4
votes
1answer
210 views

For a Noetherian scheme X, show that $X_{red}$ affine implies $X$ affine.

I have the following problem: Let X be a Noetherian Scheme and suppose that $X_{red}$ is affine. Show that this implies that X is affine. OK, so I know the "classical" proof of this using Serre's ...
8
votes
1answer
421 views

What conditions guarantee that all maximal ideals have the same height?

It fails in general that all maximal ideals in a commutative ring with unity have the same height. It's easy to construct a counter-example when the ring is NOT an integral domain (consider the ...
24
votes
5answers
3k views

Why Zariski topology?

Why in algebraic geometry we usually consider the Zariski topology on $\mathbb A^n_k$? Ultimately it seems a not very interesting topology, infact the open sets are very large and it doesn't satisfy ...