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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2
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2answers
440 views

How to define the dual sheaf

If $X$ is a scheme and a sheaf $\mathcal F$of modules on $X$, then how can we define the dual $\mathcal F^*$ of $\mathcal F$? Obviously, we set $\mathcal F^*(U)=\rm{Hom}_{\mathcal O_X(U)}(\mathcal ...
2
votes
0answers
233 views

Example on non-projectively normal variety

This question orginally comes from the exercise 3.18 of Hartshorne, Algebraic Geometry. If $Y$ is a projective variety in $\mathbb{P}^n$ then $Y$ is projectively normal (w.r.t the embedding) if its ...
4
votes
0answers
155 views

Definition of analytically unramified rings

A noetherian local ring A is said to be analytically unramified if the complete local ring $\hat{A}$ is reduced. I don't see why it makes sense to call such a ring analytically unramified. The ...
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vote
2answers
44 views

Solving for x. Do I require iterations?

I have the following expression (used in a computer program): $$f(x)=b^{{k}^{ax}}$$ where $k$ is a constant and $a$ and $b$ are given. I need to calculate the distance from this curve to a point $P: ...
6
votes
1answer
292 views

Is there an irreducible polynomial vanishing on two components? (In the Zariski sense)

The polynomial $$f(x,y) = (x^2 − 1)^2 + (y^2 − 1)^2$$ is an example of an irreducible polynomial in $\mathbf{R}[x,y]$ which is irreducible but whose zero set has multiple components in the Zariski ...
8
votes
1answer
276 views

When is the pushforward / direct image of a reflexive sheaf locally free?

I have seen a number of theorems that guarantee the direct image of a reflexive sheaf to be reflexive again, or for the direct image of a locally-free sheaf to be locally free again. This makes me ...
2
votes
0answers
56 views

restriction on the diagonal

Let $\pi_i:X\times X\to X $ be the projection of the $i$th component and $D_i (i=1,2)$ be Cartier divisors on $X$. Then is it true that $(\mathcal{O}_X(\pi_1^*D_1)\otimes\mathcal{O}_X( ...
0
votes
1answer
77 views

Discontinuity of a semialgebraic function

Let $f : \mathbb R \to \mathbb R$ be semialgebraic. Is it possible that for some $x \in \mathbb R$ the limits $f(x-)$ or $f(x+)$ does not exist? In other words can it have a discontinuity of the ...
3
votes
0answers
57 views

Is there a construction known for associating a K3 surface to a curve or cover of curves

Let $X$ be a curve of genus at least two. Then one can associate an abelian variety to $X$; this is the Jacobian. Let $X\to Y$ be a double cover of curves. Then we can associate an abelian variety to ...
5
votes
1answer
244 views

Faithfully flat morphisms with all fibers complete

Prove or disprove: if $f: X \to Y$ is faithfully flat and each fiber is complete, then $f$ is proper. (I'd especially like to see a counterexample with a morphism of finite type between varieties over ...
7
votes
1answer
281 views

Doesn't Abhyankar's lemma contradict faithful flat descent (no, but I'm confused)

Let $O_K$ be a dvr with fraction field $K$. Let $L/K$ be a tamely ramified finite Galois extension. Then, Abhyankar's Lemma implies that there exists a finite Galois extension $K^\prime/K$ such that ...
3
votes
1answer
105 views

Showing that two quadratic surface are normal

In the book "Algebraic Geometry" by Robin Hartshone, or GTM 52 for short, there is a problem of showing that two quadratic surface $Q_1: xy=zw$ and $Q_2: xy=z^2$ in $\mathbb{P}^3$ are normal. I have ...
1
vote
1answer
64 views

vanishing of higher derived structure sheaf

given a field $k$ and a proper integral scheme $f:X\rightarrow \operatorname{Spec}(k)$, is it true that $f_{*}\mathcal{O}_{X}\cong \mathcal{O}_{\operatorname{Spec}(k)}$? Consider the normalization ...
15
votes
2answers
4k views

Path to Basics in Algebraic Geometry from HS Algebra and Calculus?

In this question, Why study Algebraic Geometry?, Javier Álvarez, develops a succint but encompassing description of algebraic geometry and its spread across different areas of mathematics. Indeed, it ...
4
votes
1answer
113 views

Irreducibility preserved under étale maps?

I remember hearing about this statement once, but cannot remember where or when. If it is true i could make good use of it. Let $\pi: X \rightarrow Y$ be an étale map of (irreducible) algebraic ...
6
votes
2answers
216 views

Is the circle a rational curve and what is its function field?

It does seem like the circle ($S^1=\{X^2+Y^2=1\}\subseteq k^2$ for $k$ a field) is a rational curve: it has parameterization $X=2T/(T^2+1)$ and $Y=(T^2-1)/(T^2+1)$. On the other hand, we have a ...
7
votes
1answer
343 views

why the K3 surfaces are minimal surfaces

I need to prove that all K3 surfaces are minimal surfaces, so that every birational map between K3 surfaces is an isomorphism. I've started to read beauville's book on complex algebraic surfaces: ...
1
vote
0answers
177 views

Hilbert (polynomial) dimension and dimension of a support of a module

$\newcommand{\Supp}{\mathrm{Supp}}$ $\newcommand{\Ann}{\mathrm{Ann}}$ Let $X$ be an affine algebraic variety (over a field $K$, can assume it is algebraicaly closed), $M$ a finitely generated ...
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0answers
48 views

glueing formal sheaves to obtain a maximal ideal

consider $S=Spec(\mathbb{C}[t])$ and $C\rightarrow S$ a family of proper curves with $C_{\mathbb{C}[t,t^{-1}]}$ smooth and $C_{t=0}$ nodal given by 2 irreducible components $C_1,C_2$ that intersect ...
4
votes
1answer
129 views

Dual projective surface

Let $X \subset \mathbb{P}^3$ be a smooth surface of degree $d>1$ and consider the Gauss map $X \to \mathbb{P}^{3*}$, which sends a point of $X$ to its tangent plane. To see that the image $X^*$ of ...
6
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0answers
328 views

Rational quartic curve in $\mathbb P^3$

By using similar arguments to the ones from my answer to this question, I can prove that the homogeneous coordinate ring of the rational quartic curve in $\mathbb P^3$, that is, $$R = K[x_1, x_2, x_3, ...
5
votes
0answers
139 views

Euler characteristic of a variety and its analytification

Let $X$ be a smooth projective complex variety and $\mathcal{F}$ a coherent sheaf on $X$. Let $\tau$ be a Grothendieck topology and $$ \chi(X,\mathcal{F},\tau)=\sum_i(-1)^i ...
3
votes
1answer
676 views

The image of a morphism between affine algebraic varieties.

Suppose F is an morphism between algebraic variety V and W. Prove that the pull back F# between the coordinate ring C[W] and C[V] is surjective if and only if the morphism F is an isomorphism between ...
5
votes
1answer
114 views

Hartshorne Lemma I.6.5; Why is $\mathfrak{m}_R\cap B\neq 0$?

I've been going back through some theorems in Hartshorne's Algebraic Geometry, trying to really understand the details. I'm looking at Lemma I.6.5, which states (for those who don't have the book): ...
5
votes
1answer
318 views

Rational/meromorphic functions on a scheme

In EGA (IV, §§ 20 – 21), the sheaf of meromorphic functions $\mathscr{M}_X$ on a ringed space $(X, \mathscr{O}_X)$ is defined as the sheaf associated to the presheaf that associates to an open $U ...
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vote
0answers
40 views

Is $M_g$ a subvariety of $M_{h}$ for some $h>g$

Let $g\geq 24$. Then $M_g$ is of general type. Does there exist $h>g$ such that $M_g$ is a subvariety of $M_h$? That is, does there exist an immersion $M_g \to M_g$? If the answer is not known, ...
2
votes
0answers
33 views

Are there generalizations of Prym varieties to higher dimensions

Prym varieties are abelian varieties that are associated to a double cover of algebraic curves. Can we also associate an abelian variety to a double cover of algebraic surfaces in a reasonable way? ...
3
votes
2answers
558 views

Book recommendations for commutative algebra and algebraic number theory

Are there any books which teach commutative algebra and algebraic number theory at the same time. Many commutative algebra books contain few chapters on algebraic number theory at end. But I don't ...
2
votes
1answer
162 views

Tangent space as the dual of an ideal quotient

I'm just trying to understand better this way of seeing the tangent space. Given a manifold $M$, it's possible to define the tangent space as $(\mathfrak{I}/\mathfrak{I^2})^*$ , being $\mathfrak{I} = ...
3
votes
2answers
161 views

A vanishing theorem for differential forms.

I am trying to prove that for an algebraic surface $X$ (under some extra assumptions that are probably not important) there the space $H^0(X,\Omega_X^1)$ is trivial, i.e. that there exist no globally ...
3
votes
1answer
492 views

Normalisation of an algebraic curve.

I need to compute explicitly the normalisation of a singular algebraic curve $C$ which is given by an explicit equation in $\mathbb{A}^2$. This task is mostly reduced to finding the integral closure ...
5
votes
1answer
47 views

Is the relative rank function with respect to an ample line bundle non-decreasing

Let me make the question in the title more precise. Let $f:X\to $ Spec $k$ be a smooth projective connected variety over a field $k$ of characteristic zero. Let $\mathcal L$ be a line bundle on $X$. ...
2
votes
0answers
202 views

Diophantine equations/Diophantine Geometry

I am very knew to this site and I am eagerly waiting for solutions of: (1) Let $x$ be an algebraic number with degree $n > 1$. Then there exists only finitely many rational numbers $p/q$ (in ...
1
vote
2answers
446 views

proving that this ideal is radical or the generator is irreducible

How can I prove that the ideal $ (xy-1) \subset k[x,y] $ is radical? I think that it's enough to prove that the polynomial $xy-1$ is irreducible. How can we prove that? Here $k$ is a field, I'm not ...
7
votes
1answer
138 views

What happens if we blowup $\mathbb{P}^2$ at more than 9 points?

For $s\leq 8$, if we blowup $\mathbb{P}^2$ at $s$ general points, we get a Del Pezzo surface. I am wondering what happens if $s\geq 9$? How does this 8 being calculated?
7
votes
2answers
118 views

Explicit Resolution of Singularity

Has someone ever done/seen the explicit computation of the minimal resolution $Y$ of $X=\mathbb{C}^2/\mathbb{Z}/n= \text{Spec}\mathbb{C}[x,y,z]/(xy-z^n)?$ what is the best description of $Y$ if one ...
2
votes
1answer
200 views

proof that sum of ramification degrees is degree of morphism between curves?

If $X$, $Y$ are irreducible smooth projective curves over an algebraically closed field and $\alpha:X\rightarrow Y$ is a morphism, how do we prove that ...
6
votes
0answers
94 views

Calculation of intersection of a divisor with itself

I am reading the following paper on the Cox ring of $\overline{M}_{0,6}$ by Ana-Maria Castravet: http://arxiv.org/abs/0705.0070 I am stuck on an intersection-theoretic question, which appears as ...
2
votes
1answer
345 views

what is genus of complete intersection for: $F_1 = x_0 x_3 - x_1 x_2 , F_2 = x_0^2 + x_1^2 + x_2^2 + x_3^2$

In the below problem I want to answer second part: Show that the curve in $\mathbb{P}^3$ defined by the two equations $x_0 x_3 = x_1 x_2$ and $x_0^2 + x_1^2 + x_2^2 + x_3^2 = 0$ is a smooth complete ...
1
vote
1answer
206 views

The closed points are a constructible set?

Let $k$ be an algebraically closed field. Consider the affine scheme Spec $k[x_1,...,x_m]$. The set of closed points $k^m$(i.e. the maximal ideals) is a constructible set?
8
votes
1answer
207 views

Zariski topology analogue for non-algebraically closed fields

Let $k$ be a field and $\bar{k}$ its algebraic closure. The set $X$ of $n$-tuples over $\bar{k}$ can be given the Zariski topology in which the closed sets are the sets of zeros of sets of polynomials ...
3
votes
1answer
90 views

Can a closed subset of an affine scheme have empty interior?

I have an inclusion of closed subsets $V(J) \subset V(I)$ in an affine scheme $Spec(R)$ with the property that $V(I) = V(J) \cup \partial V(I)$. I would like to conclude that $V(J)=V(I)$. (Here ...
3
votes
1answer
162 views

An analogue of degree-genus formula for surfaces.

I have recently learnt Riemann-Roch formula for surfaces. Roughly speaking, the theorem says that on a reasonably nice surface we have the relation: $$ \chi(D) = \frac{1}{2}(D.D - D.K) + p_a + 1 $$ ...
8
votes
1answer
643 views

Automorphism group of the elliptic curve $y^2 + y = x^3$

Consider the elliptic curve $E : y^2+y = x^3$ over $\overline{\mathbb{F}_2}$. It has the biggest automorphism group $G$ among all elliptic curves, namely with order $24$. What is the structure of $G$? ...
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vote
1answer
413 views

Question on Veronese map

On page 10 of this note the author proved that the image of $\mathbb{P}^{n}$ under the Veronese map $\nu_{d}$ : $\mathbb{P}^{n}\longrightarrow \mathbb{P}^{m}$ is isomprphic to $\mathbb{P}^{n}$. In ...
5
votes
1answer
113 views

Equivalence of ample divisor

Let $X$ be a smooth complex projective variety. Having ample anticanonical class is equivalent to having $-K_{X} > 0$? How to prove that? Does it hold for any ample divisor in $X$?
2
votes
1answer
97 views

Hyperelliptic curve, injectivity of pullback homomorphism

Let $X$ be a hyperelliptic curve of genus $g$ (nonsingular, etc.) with a hyperelliptic cover $X \to \mathbb{P}^1$ corresponding to an invertible sheaf $\mathscr{L}$. The composition with the $(g-1)$ ...
2
votes
1answer
155 views

Dual curve clarification

Let $C \subset \mathbb{P}^n$ be a projective curve. What is the dual curve in general? According to Wikipedia, for a smooth hypersurface $X$ defined by a homogeneous polynomial $f,$ the dual curve ...
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vote
0answers
49 views

Are there moduli spaces of higher-dimensional varieties

In short, the answer to the question is yes. I'm aware of the existence of moduli spaces for canonically polarized varieties with fixed Hilbert polynomial over $\mathbf C$. I think they require the ...
2
votes
0answers
149 views

How to calculate the restriction of a divisor to itself?

I am wondering whether there is a notion of restriction of a divisor to itself? More specifically, let $X=\mathbb{P}^3$ be the projective space and let $x, y$ be two points in $X$, let $l$ be the line ...