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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3
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1answer
229 views

Kähler differentials not the same as regular differentials on a singular curve

Let $X$ be the affine cubic curve $y^2=x^3$, over a field of characteristic not equal to 2 or 3. Let $A=k[X]$, the ring of regular functions on $X$. Let $\Omega_A$ be the $A$-module of Kähler ...
1
vote
0answers
61 views

Finding a non-trivial regular differential form on a singular cubic surface

Let $X$ be the cubic surface $x_0^3=x_1x_2x_3$ in $\mathbb{P}^3$. How to find a non-zero regular differential 2-form on $X$?
0
votes
1answer
155 views

Closed proper subvarieties of curves are finite sets of points

Why is it true that a closed proper subvariety of a curve is a finite set of points? I had the following lines of thinking: Let $C$ be a curve and $X$ a proper closed subvariety of $C$. Write $C$ as ...
1
vote
0answers
48 views

Projection of Curve into $\mathbb{P}^{1}$.

Working out some questions from Ravi Vakil's notes. Here is a question: Question: Suppose $\operatorname{char}\bar{k} \neq 2$ and let $C$ be the curve defined by $x^{2}+y^{2} = z^{2}$. Let $\rho$ be ...
0
votes
1answer
87 views

Product of projective and affine line is not affine

Why is $\mathbb{P}^{1} \times \mathbb{A}^{1}$ not isomorphic to an affine variety?
0
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1answer
119 views

property of an affine algebraic set in Zariski topology

The following problem is giving me trouble: Suppose $X \subset \mathbb{A}^{n}$ is an affine algebraic set, and $S \subset X$ is a subset. Show that if $\bar{S}$ is the closure of $S$ in the ...
1
vote
1answer
230 views

How to find the canonical divisor on a nonsingular toric variety?

I am reading Fulton's "Toric Varieties." In it, he explains that if $X$ is a toric variety and if $D_1, \ldots, D_d$ are the irreducible divisors invariant under the big torus action, then $$ ...
4
votes
1answer
109 views

Number of birational classes of dimension d, geometric genus 0 varieties?

Fix an algebraically closed field $k$ and a positive integer $d$. My question is, what is the number of birational classes of dimension $d$, projective varieties over $k$ with geometric genus 0? If it ...
7
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0answers
475 views

A problem about the twisted cubic

I have some difficulty with the following problem: Let $f : k → k^3$ be the map which associates $(t, t^2, t^3)$ to $t$ and let $C$ be the image of $f$ (the twisted cubic). Show that $C$ is an ...
6
votes
1answer
125 views

Diagonal in projective space

This is exercise $2.15$ from Harris book "Algebraic Geometry: A First Course". Show that the image of the diagonal in $\mathbb{P}^{n} \times \mathbb{P}^{n}$ under the Segre map is isomorphic to the ...
31
votes
2answers
1k views

How do different definitions of “degree” coincide?

I've recently read about a number of different notions of "degree." Reading over Javier Álvarez' excellent answer for the thousandth time finally prompted me to ask this question: How exactly do ...
10
votes
1answer
308 views

Problem about Complete Intersection in $\textbf P^n$ (from Hartshorne).

I am in trouble with Exercise 8.4 in Hartshorne's Chapter II; I am doing part (a). It is about (global) complete intersection in $\textbf P^n$. For those without Hartshorne' book at hand, I describe ...
8
votes
3answers
322 views

Krull Dimension of a scheme

Can someone give a hint or a solution for showing that a scheme has Krull dimension $d$ if and only if there is an affine open cover of the scheme such that the Krull dimension of each affine scheme ...
4
votes
1answer
322 views

Ramification on hyperelliptic curves

I am using Rick Miranda's book "Algebraic curves and Riemann Surfaces" to try and check some things about hyperelliptic curves. I have completed almost all of one of the exercises, but there is one ...
3
votes
1answer
143 views

The probability of $Ax^2+Bxy+Cy^2 = 1$ defining an ellipse.

In Keith Kendig's paper, Stalking the Wild Ellipse (published in the American Mathematical Monthly, November 1995), he says that if $A, B, C$ are chosen at random, the probability that the Cartesian ...
8
votes
2answers
237 views

connections on coherent sheaves

Let $X$ be a smooth variety over $\mathbb{C}$. If $\mathscr{F}$ is a coherent sheaf on $X$ with connection, does it follow that $\mathscr{F}$ is locally free? I can't think of any counterexamples. ...
7
votes
2answers
270 views

Is this quotient ring $\mathbb{C}[z_{ij}]/\ker\phi$ integrally closed?

A few days ago, I asked a linear algebra question, but it seems that the notions are better stated in terms of algebraic geometry. I don't have much solid knowledge of algebraic geometry, so I'm ...
2
votes
2answers
189 views

What if $\operatorname{char}\mathbb{K}$ is not $0$ or if $\mathbb{K}$ is not algebraically closed? (Nullstellensatz)

Given a field $\mathbb{K}$ which is algebraically closed and of characteristic 0, we can say exactly what the maximal ideals of $\mathbb{K}[x_1,\dots,x_n]$ are and they correspond to points in ...
0
votes
1answer
91 views

Image of $2$-fold map

Show that the image of the veronese map $[a : b] \mapsto [a^{2}: b^{2} : ab]$ is not contained in any hyperplane of $\mathbb{P}^{2}$. Using the result from a previous question I asked: Hypersurface ...
2
votes
2answers
336 views

What is Hilbert polynomial of this projective variety?

Suppose you have a map $\varphi\colon\mathbb{C}^m\times\mathbb{C}^n\to\mathrm{Mat}_{m,n}(\mathbb{C})$ defined by sending $(\mathbf{u},\mathbf{v})\mapsto\mathbf{u}\cdot\mathbf{v}^T=(u_i,v_j)$. So ...
3
votes
1answer
255 views

Hypersurface becomes an hyperplane after embedding

Let $X$ be an hypersurface of degree $k$ in $\mathbb{P}^{n}$, why the equation defining $X$ becomes linear in the Veronese coordinates? More precisely I want to understand the last paragraph of the ...
4
votes
1answer
259 views

Resolution of singularities (small resolution)

Consider the following complex (complete intersection) variety, $$ f_1: x_0^2 + x_1^2 + x_2^2 + x_3^2 = 4x_4x_5,$$ $$ f_2: x_4^4 + x_5^4 = 2x_0x_1x_2x_3,$$ in $\mathbb{P}^5$. This is the first example ...
3
votes
1answer
184 views

Explicit example of a toric flip

I am looking for a toy example of a flip between toric projective 3-folds. More precisely, I would like to see their defining fans (or polytopes). Does anyone know where I can find something like ...
6
votes
1answer
431 views

Why morphism between curves is finite?

If $X$ is a complete nonsingular curve over $k$, $Y$ is any curve over $k$, $f: X \to Y$ is a morphism not map to a point (so $f(X)=Y$), then $f$ is a finite morphism. This is the assertion prove in ...
2
votes
0answers
301 views

Elliptic curves, 2-torsion and branch points.

I'm currently reading through Ravi Vakil's notes on Algebraic Geometry. I've been having trouble grasping some things conceptually though and I hope that you can help me. For an elliptic curve (E,p) ...
4
votes
1answer
98 views

How to get a geometric morphism out of a section? (And general pedagogy on classifying toposes)

Let $\mathcal{E}$ and $\mathcal{F}$ be toposes, $X$ an object of $\mathcal{E}$ and $p: \mathcal{E}/X \rightarrow \mathcal{E}$ the canonical geometric morphism (whose inverse image part is pullback ...
11
votes
2answers
911 views

Why is Hodge more difficult than Tate?

There are strong connections between the Hodge and the Tate conjectures, mainly at the level of similarities and analogies. To quote from an answer of Matthew Emerton on MathOverflow: "[...] we ...
3
votes
0answers
88 views

Closed / embedded surface

Given a closed surface in $\mathbb R^3$, is it necessarily an "embedded surface"? I think it is true, but that is just because I can't think of a closed surface for which we cannot construct a smooth ...
7
votes
1answer
192 views

Why is the kernel of this strange polynomial homomorphism what it is?

I've been trying to delve a little further into linear algebra, but I'm not following something I think is supposed to be obvious. Suppose $M_{m,n}(\mathbb{C})$ is the set of rectangular $m\times n$ ...
3
votes
1answer
110 views

Varieties given by non-algebraic equations

In algebraic geometry one (mostly) studies varieties given by polynomial equations. Such equations define algebraic varieties and there are many "dictionaries" available. For example, the category ...
4
votes
0answers
67 views

Automorphism of $L|K$ mapping 3 distinct rational points of $S_{L|K}$ to other 3 distinct ones

Let $K$ be a field and consider $L = K(x)$ the field of rational functions. Let $v_{1}, v_{2}, v_{3}$ rational points in the abstract Riemann surface $S_{L|K}$, distinct from each other, and $w_{1}, ...
9
votes
1answer
692 views

Finding singular points and computing dimension of tangent spaces (only for the brave)

I'm currently looking at the following two questions: i) Consider $V = Z(I) \subset \mathbb A_k^3$ where $I$ is generated by $X_1^3 - X_3$ and $X_2^2-X_3$. Find the points at which $V$ is singular ...
5
votes
1answer
224 views

Finding the radical of some ideals

I need to find the radicals of the following ideals: i) $\mathfrak{a} = (xy^3, x(x-y))$ ii) $\mathfrak{b} = (xy^3, x^2(y-3))$ iii) $\mathfrak{c} = (x^2(y-z), xy(y-z), xz(y-z)^2)$ Can I just use ...
2
votes
2answers
176 views

Components of a variety in projective space

How do we find the irreducible components of the following projective variety in $\mathbb{P}^{3}$, $V(wy-x^{2},xz-y^{2})$?
3
votes
1answer
499 views

Function fields of irreducible varieties

The following might seem long, rambling and to contain more information than necessary. The problem is, I'm having a macro-understanding issue and feel like I need to tell you everything I think so ...
2
votes
1answer
80 views

Representation of a subset of a finite affine space as a variety

It is simple to see that every subset of a an affine space over a finite field is a variety - for example, it follows from the fact that finite subsets are closed in the Zariski Topology of every ...
0
votes
0answers
61 views

Continuity of a map of a topological space to a pro-topological space

Let $(X_i)$ be a projective system of topological spaces. Let $X$ be the projective limit of $X_i$. Let $G$ be a topological space. What does it mean for $G\to X$ to be continuous? My guess is that ...
1
vote
0answers
271 views

Artin-Schreier extensions over characteristic two fields

I have been looking at hyperelliptic curves over an algebraically closed field $k$ of characteristic two, with a view towards finding the basis for the vector space of holomorphic differentials. To do ...
1
vote
3answers
143 views

Intersection of projective curves

In general, how do we find the intersection of projective curves? For example suppose I have $V_{1}(x^{2}+y^{2}-2z^{2}),V_{2}(x^{2}+y^{2}-z^{2})$ and I want to find $V_{1} \cap V_{2}$ viewed as ...
1
vote
2answers
83 views

If $Y = X \backslash Z(f)$ for some affine variety $X$ with $p \in Y$, then $T_p Y \cong T_p X$

Let $Y = X \backslash Z(f)$ be a quasi-affine variety for some affine variety $X$, and let $p \in Y$. I'd like to prove that $T_p Y \cong T_p X$. I have the following definition of $T_p X$: If ...
2
votes
1answer
257 views

Why are $\mathbb A_k^2 \backslash \{(0,0) \} $ and $\mathbb P_k^2 \backslash \{(0,0) \} $ not isomorphic to affine nor projective varieties?

Why are $\mathbb A_k^2 \backslash \{(0,0) \} $ and $\mathbb P_k^2 \backslash \{(0,0) \} $ isomorphic to neither affine nor projective varieties? I've seen this question in several different places, ...
3
votes
1answer
80 views

Twists of rational points

Let $X$ be a "nice" algebraic curve over some field $K$, say characteristic zero. The twists of $X$, i.e., the curves $Y$ over $K$ such that $X_{\overline K} \cong Y_{\overline K}$, are in bijection ...
4
votes
1answer
77 views

Product of affine schemes

For any ring $A$, define a functor $\text{Spec}(A)$ from rings to sets by $$\text{Spec}(A)(R) = Hom_{\text{Rings}}(A,R)$$ Call a functor $X$ an affine scheme if it is isomorphic to a functor of the ...
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vote
0answers
66 views

Calculating germs for complete holomorphic function

I'm trying to find the germs $[z,f]$ for the complete holomorphic function $\sqrt{1+\sqrt{z}}$ . The question indicates that I should find 2 germs for z = 1, but I seem to be able to find 3! Where ...
4
votes
1answer
198 views

Jacobian of a curve

Let $C$ be a curve and $J$ be its Jacobian. What is the relation between $H^1(C,\mathcal{O}_C)$ and $H^1(J,\mathcal{O}_J)$ ? Can someone point me to an easy reference for this subject?
9
votes
1answer
433 views

How to tell if an element of a quotient ring is a zero divisor

I am looking at Hartshorne Example III.9.8.4., p260. He says that $a$ is not a zero divisor in $k[a,x,y,z]/I$, where $$ I = (a^2(x+1) -z^2, ax(x+1)-yz, xz-ay,y^2-x^2(x+1)). $$ Is there a good way to ...
4
votes
2answers
264 views

references for singularities: does quotient singularities imply gorenstein?

Is there a good place where to learn about singularities of algebraic varieties? OK, this question is terribly vague, so I'll ask what I'm currently interested in: if X is a smooth variety and G is a ...
4
votes
2answers
206 views

Reference request for “Hodge Theorem”

I have been told about a theorem (it was called Hodge Theorem), which states the following isomorphism: $H^q(X, E) \simeq Ker(\Delta^q).$ Where $X$ is a Kähler Manifold, $E$ an Hermitian vector ...
2
votes
2answers
40 views

Showing that a set is closed in the base variety of a family $V\to B$.

We work over a field $k$. Let $B$ be an algebraic variety over $k$. Suppose we are given a family of subvarieties of $\textbf P_k^n$ with base $B$, by which I mean a subvariety $V\subset ...
4
votes
0answers
259 views

Topological definition of intersection multiplicities of algebraic varieties

Let $X$ be an algebraic variety over the field of complex numbers. In other words, $X$ is a reduced separable scheme of finite type over the field of complex numbers. Let $U$ and $V$ be irreducible ...