Tagged Questions

5
votes
1answer
57 views

Degree of a Cartier Divisor under pullback

This is question 7.2.3 in Liu's book Algebraic Geometry and Arithmetic Curves and I have been trying with this for some time now. Let $f:X \rightarrow Y$ be a morphism of Noetherian schemes, and ...
8
votes
0answers
84 views
+50

Prop. 3.2.15 in Liu: Geometrically reduced algebraic variety

I have a problem with proposition 3.2.15 of Algebraic geometry and arithmetic curves of Qing Liu: Let $X$ be an integral algebraic variety over $k$. If $X$ is geometrically reduced then $K(X)$ is a ...
3
votes
1answer
33 views

Is the height associated to a degree zero divisor always bounded?

Let $X$ be a smooth, projective, geometrically irreducible curve defined over a number field, and let $D$ a divisor on $X$. To these data, we can associate a height function on the ...
2
votes
0answers
60 views

Pole of differential

Let $E : y^2 = x^3 + ax + b$ be an elliptic curve over the field $K$, char $K \ne 0$. We know that the differential $\omega = \frac{dx}{y}$ is holomorphic in infinity because we can write it as ...
8
votes
1answer
82 views

Varieties with the property that the cotangent bundle restricted to a complete nonsingular curve is free

Let $X$ be a $d$-dimensional smooth projective connected variety with cotangent sheaf $\Omega^1_X$ over $\mathbb C$. Suppose that for any nonsingular complete curve $C$ and non-constant morphism ...
8
votes
2answers
130 views

Intuitively, what is the height of a point on an abelian variety?

I have been reading through Silverman's classic text on elliptic curves and I just can't seem to wrap my head around the height functions. It just kind of shows up. What exactly does the height ...
6
votes
1answer
82 views

Are varieties of Kodaira dimension zero precisely the varieties with torsion canonical sheaf

Let $B$ a smooth projective connected variety over $\mathbf C$. Suppose that $K_B$ is torsion. Then, clearly, the Kodaira dimension of $B$ is zero. Does the converse hold? That is, suppose that $B$ ...
7
votes
2answers
108 views

Are endomorphisms of degree one always automorphisms

Let $B$ be a smooth projective connected variety over $\mathbb C$. Let $\sigma:B\to B$ be an endomorphism of degree one. Do I understand correctly that $\sigma$ is an automorphism? I believe this ...
5
votes
1answer
50 views

Why do varieties with torsion canonical sheaf have finite etale covers with trivial canonical sheaf

Let $B$ be a variety with torsion canonical sheaf, i.e., $\omega^{\otimes n}_B \cong \mathcal O_B$ for some $n>0$. Then, why does there exist a finite (etale?) morphism $X\to B$ such that $K_X$ is ...
6
votes
0answers
46 views

Do K3 surfaces with an Enriques involution have a polarization of bounded degree

Does there exists a real number $C$ with the following property. For any Enriques surface $E$ over a number field $K$ with K3 cover $X\to E$, there exists an ample divisor $H$ on $X$ such that $H^2 ...
3
votes
1answer
43 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 ...
3
votes
1answer
40 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
1answer
121 views

Picard group of genus one curve

Is there a known example (or at least moral reason why such a thing should exist) of a genus $1$ curve $C/k$ over a field (assume perfect if you want) with no rational points such that ...
2
votes
1answer
34 views

Varieties with infinitely many topological covers of finite degree

Let $X$ be a smooth projective connected variety over $\mathbf C$ with infinitely many etale covers. If $\dim X =1$, this holds if and only if the genus of $X$ is positive. Do we have a similar ...
4
votes
0answers
50 views

Why should automorphism groups of compact hyperbolic curves be finite

Let $X$ be a compact connected Riemann surface of genus at least two, or let $X$ be a smooth projective connected curve over an algebraically closed field of characateristic zero. Then Hurwitz proved ...
3
votes
0answers
31 views

What are the easiest surfaces of general type

The "easiest type" of curves of general type are those of genus two. In this case $\chi(X,\mathcal O_X) = -1$ and $\deg K = 2$, where $K$ is the canonical sheaf. I'm a bit lost when it comes to ...
5
votes
1answer
69 views

Ramification indices and residue degrees of a finite Galois extension

Let $A$ be a dvr with fraction field $K$ of characteristic zero. Let $L/K$ be a finite Galois extension and let $B$ be the integral closure of $A$ in $L$. For a prime $b$ of $B$, let $e_b$ be its ...
5
votes
1answer
78 views

Definition of tamely ramified

I think I can show that the following definitions of "tamely ramified" coincide. I thought it would be good to be sure. Sorry for the easy questions. Let $O_K$ be a dvr with maximal ideal $\mathfrak ...
9
votes
2answers
129 views

Elliptic curves over Spec Z

I want to show that there are only finitely many elliptic curves over Spec $\mathbf Z$ without appealing to Siegel's theorem or Shafarevich' theorem. Firstly, I think (but I am not sure) that such an ...
2
votes
0answers
32 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 ...
6
votes
1answer
134 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 ...
1
vote
0answers
35 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 ...
1
vote
0answers
36 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
1answer
45 views

Pulling-back a divisor and reducing it

Let $f:C\to B$ be a finite morphism of curves. Let $D$ be a divisor on $B$. Does the equality of divisors $$(f^\ast D)_{red} = f^\ast (D_{red})$$ hold on $C$? (I'm asking for an equality of divisors, ...
8
votes
0answers
128 views

Self-Intersection Number $-2$

I am new here, but hopefully you can help me with a concrete problem I have. I try to compute a Self-Intersection Number of a constructed curve in an analytic surface. I know the answer by some ...
7
votes
0answers
102 views

Where does this elliptic curve come from?

In Zeta functions of an infinite family of K3 surfaces, Scott Alhgren, Ken Ono and David Penniston compute the zeta functions (given a good reduction restriction mentioned below) of the K3 surfaces ...
4
votes
1answer
59 views

Why is the rank of $f_\ast L$ the degree of $f$

Let $f:X\to Y$ be a finite morphism of curves. Let $L$ be a line bundle on $X$. Why is $f_\ast L$ a line bundle and is the degree of $f_\ast L$ equal to $\deg f$ or $\deg f+ \deg L$? Here is my ...
4
votes
0answers
114 views

If $X\times C$ is isomorphic to $Y\times C$, does it follow that $X$ is isomorphic to $Y$

Let $k$ be a field. Let $X$, $Y$ and $C$ be varieties over $k$ such that $X\times_k C $ is isomorphic to $Y\times_k C$. Assume that $C$ is a curve. (The varieties $X$, $Y$ and $C$ are assumed to be ...
2
votes
1answer
72 views

Abelian subvarieties of a principally polarized abelian variety are principally polarized

Let $A$ be a principally polarized abelian variety. Let $X\subset A$ be an abelian subvariety. Is $X$ also principally polarized? Here's what I think should be a proof. Is it correct? We may and do ...
6
votes
1answer
119 views

Are there infinitely many pairs of rational numbers such that…

Are there infinitely many pairs of rational numbers $(a,b)$ such that $a^3+1$ is not a square in $\mathbf{Q}$, $b^3+2$ is not a square in $\mathbf{Q}$ and $b^3+2 = x^2(a^3+1)$ for some $x$ in ...
4
votes
1answer
54 views

The canonical divisor of the projective line

Let $A$ be a ring. Assume it has some nice properties if necessary, e.g., $A$ is a Dedekind domain. Let $\mathbf{P}^1_A$ be the projective line over $A$. I want to show that $-2 [\infty]$ is a ...
2
votes
1answer
73 views

Why do number rings have no endomorphisms

This question is about the analogy between number fields and function fields. It's a soft question and the title misrepresents the question. Consider the projective line over a field. This has many ...
2
votes
0answers
75 views

Do K3-surfaces have Weierstrass equations

I've been wondering a bit about K3-surfaces and their analogy to elliptic curves. I've just started so this might be a very silly question. Do all K3-surfaces have a Weierstrass equation (up to ...
2
votes
1answer
80 views

Primes of good reduction for varieties

Suppose I have a (smooth, projective) variety $X/\mathbb{Q}$. Is the notion of a primes of bad/good reduction well-defined from this data? Motivation and attempt at an answer: The question should be ...
1
vote
2answers
52 views

Factorizing rational functions of curves

Let $f:X\to \mathbf{P}^1$ be a rational function of degree $d\geq 2$ on a curve $X$. Let $n\geq 2$ be a divisor of $d$. Does there exist a curve $Y$ with a rational function $g:Y\to \mathbf{P}^1$ of ...
3
votes
1answer
64 views

Does de Franchis' theorem hold over any base field

Let $k$ be a field and let $X$ be a hyperbolic curve over $k$. Then, there are only finitely many hyperbolic curves $Y$ over $k$ dominated by $X$. I know this statement holds over $k=\mathbf{C}$. In ...
0
votes
1answer
56 views

Representing a curve as a plane curve in different ways

Let $X$ be a (smooth projective geometrically connected) curve over $k$, where $k$ is a field of characteristic zero. We assume $X$ has genus at least $2$. I know that $X$ has a plane model. More ...
5
votes
1answer
77 views

The number of curves of given genus over a field

Let $k$ be a field. Let $g\geq 0$ be an integer. I have an elementary question. Let $N$ be the "number" of $k$-isomorphism classes of smooth projective geometrically connected curves over $k$ of ...
2
votes
1answer
85 views

the elliptic curves with j-invariant zero

Let $B\in K^\ast$, where $K$ is a number field. Let $y^2=X^3+B$ be the Weierstrass equation for an elliptic curve $E_B$ over $K$. Note that the $j$-invariant of $E$ is zero. When is $E_B$ ...
3
votes
1answer
60 views

How can function fields have different degrees over the projective line

I'm confused. Let $X$ be a curve over a field $k$. Let $K= K(X)$ be its function field. Then, $K(X)$ is a field. Each non-constant morphism $f:X\to \mathbf{P}^1_k$ gives a field extension ...
2
votes
1answer
100 views

writing down the minimal discriminant of an elliptic curve

Let $j$ be an integer. Does there exist an elliptic curve $E_j$ over $\mathbf{Q}$ with $j$-invariant equal to $j$ whose minimal discriminant we can write down in a practical way? For example, can ...
5
votes
1answer
167 views

Conditions for a curve to be defined over a subfield

I have just finished reading Hartshorne, Chapter 1, Section 6 and have some questions about curves defined over a subfield of an algebraically closed field. For simplicity, let $k$ be a perfect field, ...
16
votes
0answers
166 views

Tate conjecture for Fermat varieties

I've been looking at Tate's Algebraic Cycles and Poles of Zeta Functions (hard to find online... Google books outline here) and have a question about his work on (conjecturing!) the Tate conjecture ...
2
votes
1answer
105 views

Can we descend field extensions of prime degree of number fields to number fields of the same degree

Let $K$ be a number field and let $p$ be a prime number. Let $L$ be a degree $p$ field extension of $K$. Does there exist a degree $p$ field extension $M$ of $\mathbf{Q}$ such that ...
3
votes
3answers
153 views

How do I write down a curve with exactly one rational point

Let $g\geq 1$. I would like to write down (for all $g$) a smooth projective geometrically connected curve $X$ over $\mathbf{Q}$ of genus $g$ with precisely one rational point. Is this possible? For ...
4
votes
2answers
114 views

Genus of curves embedded into some projective space

The Plucker formula allows one to calculate the genus of a curve embedded into $\mathbf{P}^2$. Does there exist a "higher-dimensional" analogue of the Plucker formula? More precisely, let ...
2
votes
2answers
123 views

Rational points on singular curves and their normalization

Let $X$ be a curve over a field $k$. Assume that $X$ is geometrically connected, geometrically reduced and stable. Let $Y\to X$ be the normalization. Is $Y(k) = X(k)$?
2
votes
2answers
87 views

genus of normalization of stable curve

Let $X$ be a stable curve of genus $g$ over the field $k$, i.e., a $k$-rational point of the Deligne-Mumford stack $M_g$. What is the genus of the normalization of $X$? Does it depend on the number ...
0
votes
2answers
125 views

Why are these curves not defined over a smaller field

Let $K$ be a number field and let $\pi$ be an element in $K$. Assume that $\pi$ is not contained in a subfield of $K$. Consider the curve $y^2 = x^{2g+1}+\pi$. This defines (after homogenization and ...
3
votes
1answer
69 views

When is this quotient by an action on the product of a variety with itself non-singular

Let $X$ be a smooth projective geometrically connected variety over a field $k$. Let the cyclic group $G=\{e,a\}$ with two elements act on $X \times X$ via $a\cdot (x_1,x_2) = (x_2,x_1)$. When is ...

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