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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 ...
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 ...
8
votes
0answers
76 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 ...
8
votes
0answers
127 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 ...
7
votes
0answers
160 views
Do Neron models of hyperbolic curves exist
Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$.
Does there exist a Neron model $\mathcal X$ for $X$ over $O_K$?
By a Neron model, I mean ...
7
votes
0answers
62 views
Which number fields allow higher genus curves with everywhere good reduction
The field of rational numbers is not such a number field. That is, there does not exist a smooth projective morphism $X\to\mathrm{Spec} \mathbf{Z}$ such that the generic fibre is a curve of genus ...
6
votes
0answers
51 views
Families of curves over number fields
Is there known a number field $K$ and a curve $F(x, y) \in K(t)[x, y]$ such that $F(x, y)$ does not have points over the field of rational functions $K(t)$ but for all but finitely many positive ...
6
votes
0answers
45 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 ...
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 ...
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 ...
4
votes
0answers
69 views
Curve - non singular curve and its genus
Help me please with this problem:$ X \subset \mathbb{P}^{2}$
defined as $x^{3}y+y^{3}z+z^{3}x=0$
1.Prove X - non singular curve and find its genus.
2.Prove X - maximal curve over $F_{8}$ field, and ...
4
votes
0answers
75 views
On local rings of a normalization
Let X an irreducible singular curve with a singular point $x$. Consider $A$, the normalization of $\mathcal{O}_{X,x}$, and $x_1,\dots,x_r$ the points over $x$ in the normalization of $X$. Why $A\cong ...
4
votes
0answers
78 views
find valuations
consider $f(x)=x^4+5x^3+1$. Let $\alpha$ be a root of this polynomial and consider $K=\mathbb{Q}(\alpha)$. Which are the absolute values on $K$ extending the usual absolute values on $\mathbb{Q}$? Are ...
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 ...
3
votes
0answers
43 views
What applications does the theory of fibered surfaces have
Let $C$ be a smooth projective connected curve over $\mathbf{C}$.
Let $X$ be a curve over the function field of $C$.
Arakelov and Parshin proved the Mordell conjecture by considering a model for $X$ ...
3
votes
0answers
68 views
Does composing the Frobenius with an automorphism give another Frobenius
Let $X_0$ be a variety over $\mathbf{F}_q$. Consider the Frobenius $F_0:X_0\to X_0$. Let $X= X_0\times \bar{\mathbf{F}_q}$ and let $F:X\to X$ be $F_0 \times \textrm{id}$.
Let $f:X\to X$ be an ...
3
votes
0answers
92 views
places valuations and so on
I am looking for complete references on places, valuations and so on. In particular I need to understand the meaning of "a finite place does not divide a prime number $\ell$" and what is the Frobenius ...
2
votes
0answers
58 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 ...
2
votes
0answers
31 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 ...
2
votes
0answers
71 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
0answers
45 views
What are the branch points of $X(n)\to X(1)$
Let $\Gamma \subset \mathrm{SL}_2(\mathbf{Z})$ be a finite index subgroup. Let $X_\Gamma \to X(1)$ be the corresponding morphism of compact connected Riemann surfaces (obtained by adding the cusps).
...
2
votes
0answers
63 views
Is the degree of a Galois morphism bounded by $84(g-1)$
Let $X\to Y$ be a finite morphism which is Galois in the category of smooth projective connected curves over $\mathbf{C}$.
Assume $g=g(X) \geq 2$.
Is the degree of $X\to Y$ bounded by $84(g-1)$?
I ...
2
votes
0answers
70 views
Why is the trace map on an abelian variety continuous
Let $X$ be a (EDIT) variety with a group structure. For $a\in A$, let $t_a$ be the translation on $X$: $t_a(x) = a+x$.
Why is the function $f:X\to \mathbf{C}$ given by $$f(a) = \sum_{i} (-1)^i ...
2
votes
0answers
130 views
Minimal resolution of singularities of Fermat curve
Fix a prime number $p$.
Let $F$ be the geometrically connected Fermat curve of exponent $p$ over a number field $K$, i.e., $F$ is given by the equation $x^p+y^p = z^p$. Consider the same equation ...
2
votes
0answers
101 views
Primitive nth roots of unity and moduli of elliptic curves
In various descriptions of the moduli space of elliptic curves with level structures, such as the description of $X_0(N)$ being defined over $Spec\ \mathbb{Z}[1/N]$, the primitive $N^{th}$ roots of ...
1
vote
0answers
45 views
Galois invariant of Tate twists
let $k$ be the maximal extension of $\mathbb{Q}$ unramified outside a set $T$ of primes in $\mathbb{Z}$.
Take a $p\in T$ and set $G=Gal(k/\mathbb{Q})$. I would like to now if there is a classical ...
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 ...
1
vote
0answers
30 views
CAS for counting points of varieties over finite fields
I am looking for a computer algebra system, which is able to some of the following (in theory equivalent) things for a smooth projective variety defined over a finite field:
-Count the number of ...
1
vote
0answers
116 views
units in discrete valuation rings
Let $B/A$ be an extension of discrete valuation rings such that the purely inseparable extension of residue fields $l/k$ has a primitive element, i.e., $l=k(y)$ for some element $y$ in $l$. I want to ...
1
vote
0answers
40 views
action of tori on singular jacobian
I have the 2 following presentations:
assume $G$ is a semiabelian variety, i.e. an extension of an abelian variety by a torus $0\rightarrow T\rightarrow G\rightarrow A\rightarrow 0$. It is ...
0
votes
0answers
59 views
invert Grothendieck spectral sequence
I have 4 topoi $A,B,C,D$ (these are associated to abelian sheaves on some sites) and functors (which corresponds to some push-forward of sheaves)
$F: A\rightarrow B$
$G: B \rightarrow C $
$H: A ...
0
votes
0answers
96 views
What is the Hurwitz number of an elliptic curve
One can associate a Hurwitz number to any rational function $f:X\to \mathbf{P}^1$ on a compact connected Riemann surface $X$ which ramifies over precisely FOUR points.
Suppose that $X$ is an elliptic ...
0
votes
0answers
56 views
What is the moduli space of smooth AFFINE curves of given genus
It is interesting to study the moduli space $M_g$ of smooth projective curves of genus $g$.
Why not smooth affine curves of genus $g$?
0
votes
0answers
154 views
a question on the Poincaré bundle
Let $C$ be a smooth curve. Letting $J$ be its Jacobian, consider the Poincaré bundle $\mathcal P$ on $J\times J$. Let $p: J\times J\rightarrow J$ be the projection.
How can I compute the complex $R ...
0
votes
0answers
40 views
extend a morphism knowing it on etale covering
Let $S$ be a separated scheme. Let $U,J$ be separated schemes over $S$. Assume we can construct, after etale base change $T\rightarrow S$, a map $U_T\rightarrow J_T$ (the sub-T indicates the pulled ...
0
votes
0answers
53 views
cohen-macaulay implication
why a torsion free flat module over a discrete valuation ring is Cohen-Macaulay?
0
votes
0answers
120 views
Normalization of a node
how it is described the normalization of $Spec(k[[x,y]]/(xy))$? In particular call this $N$. Why $N$ can be written as the inersection of 2 local rings and the same holds for the maximal ideal (with 2 ...
