The ramification tag has no wiki summary.
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Can ramification be “seen” at smooth subcurves?
This is a follow-up question to this. Though it's a differnt question, therefore I decided to open a new question so that we don't lose track. I hope that's okay.
Let $f: X \to Y$ be a finite, ...
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Control ramification in Noether Normalization
Let $X$ be an irreducible affine variety (integral, reduced scheme of finite type over an algebraically closed field $\Bbbk$ of characteristic zero) of dimension $n$. The well-known Noether ...
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When does a dyadic prime ramify in a relative quadratic extension?
In a quadratic extension $\mathbb{Q}(\sqrt{d})$of $\mathbb{Q}$ it is clear that 2 ramifies if and only if $d\equiv 2,3\mod 4$ (easy to see if you compute the discriminant). But if I take a relative ...
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Degeneracy divisor of the “trace” morphism
Let $f\colon X\to Y$ be a finite morphism of smooth curves over an alg. closed field of characteristic zero. I recently asked how methods reminiscent of basic algebraic number theory can be used to ...
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1answer
283 views
Showing that $2c_1(f_*\mathscr O_X)=-f_*R_f$ on curves, maybe by local fields
I originally asked this question on Mathematic StackExchange, but it did not seem to be attracting any attention, so now I am trying mathoverflow. I hope it is not too simple or unappropriate a ...
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1answer
248 views
Root discriminant lower bounds in algebraic geometry
Let $X$ be a simply-connected smooth projective variety over $\mathbb C$. Let $C$ be a curve on $X$.
If $Y$ is a ramified cover of $X$ of degree $n$, and $D$ is the branch divisor of $Y$, call $(D ...
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2answers
368 views
How to explicitly see the ramification over infinity
Take the equation $y^{d}=\Pi_{1}^{n}(x-t_{i})^{m_{i}}$ over $\mathbb{C}$. This affine equation gives a cyclic cover of $\mathbb{P}^{1}$. Now it is usually said without explanation that if the sum ...
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246 views
Basic arithmetic behind ramification in quadratic number fields
Ramification of prime numbers in number fields is a topic relevant to what I'm studying (arithmetic hyperbolic 3-manifolds), and many results from algebraic number theory are used there, however most ...
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137 views
ramification of discrete valuation field
Let $K$ be a discrete valuation field with valuation $v:K\rightarrow \mathbb Z\cup \{\infty\}$ which is normalized by $v(\pi)=1$ for a prime element $\pi$. Let $v:\overline K\rightarrow \mathbb ...
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Two different definitions of $\sigma$-L-spaces in Kottwitz I and II
In his papers "Isocrystals with additional structure" I and II, Kottwitz defines the notion of $\sigma$-$L$-spaces. In the first one the situation is the following
$k$ an algebraically closed field ...
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352 views
ramified quaternion algebras
I'm trying to better understand the connection between the concepts of ramification of a field extension, and ramification of a quaternion algebra. I'm also trying to build a better understanding of ...
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Question about local description of the branch locus
Let $\pi:Y\to X$ be a dominant, finite morphism of nonsingular varieties over an algebraically closed field $\Bbbk$. Assume furthermore that for all $Q\in Y$, with $P=\pi(Q)$, we have
$$\mathcal ...
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Elementary proof of the Hurwitz formula
I am aware of two forms of the Hurwitz formula. The first is more common, and deals only with the degrees. So if $f:X \rightarrow Y$ is a non-constant map of degree $n$ between two projective ...
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150 views
Unicritical rational functions on curves in characteristic $p$
Let $k$ be an algebraically closed field of positive characteristic $p$, and let $X_{/k}$ be a smooth projective connected curve. Let $x_0$ be a point of $X(k)$.
How precisely can one describe ...
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258 views
analog for the discriminant of number fields in the function field case?
Is there a nice algebraic way of determining the ramification of a morphism between curves? Ie, some analog of the discriminant of number fields?
Specifically, I'm trying to prove that if $X$ is a ...
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2answers
445 views
trying to understand the support of the sheaf of relative differentials
So I'm trying to understand a proof of Belyi's theorem from http://eprints.soton.ac.uk/29785/1/b45h1koe.pdf
specifically lemma 3.4.
The setup is as follows: Let $X/\mathbb{C}$ be a curve, and let $t ...
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534 views
Fibre cardinality of an unramified morphism
Let $\varphi: X \to Y$ be a finite, dominant, unramified morphism of varieties over an algebraically closed field. If necessary, we can assume $X$ and $Y$ to be nonsingular. I am trying to prove that
...
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Finite, Étale Morphism Of Varieties
I have a, probably very simple, question: My intuition tells me that the following statement should be true, but I couldn't find it anywhere and I wanted to make sure I am not missing something.
Let ...
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4answers
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Higher dimensional version of the Hurwitz formula?
In Hartshorne IV.2, notions related to ramification and branching are introduced, but only for curves. The main result is the Hurwitz formula.
Now if you have a finite surjective morphism between ...
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2answers
635 views
Ramification divisor associated to a cover of a regular scheme
Let $S$ be the spectrum of $\mathbf{Z}$ or the spectrum of an algebraically closed field. (Actually, one can take $S$ to be any noetherian integral regular scheme.)
Let $f:X\longrightarrow Y$ be a ...
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1answer
556 views
Ramification in p-division fields associated to elliptic curves with good ordinary reduction
Dear MO,
Let $p$ be a prime and let $E/\mathbb{Q}_p$ be an elliptic curve. Suppose that $E/\mathbb{Q}_p$ has good ordinary reduction at $p$. In his 1972 paper ``Propriétés galoisiennes des points ...
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329 views
Can we construct rational functions with prescribed ramification on an algebraic curve over \Qbar
Let $C$ be a smooth projective connected curve of genus $g$ over $\bar{\mathbf{Q}}$. Fix a finite non-empty (Edit) set of closed points $S$ in $C$ and let $U$ be the complement of $S$ in $C$.
Q1. ...
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415 views
Does combining Abhyankar's Lemma and embedded resolution give horizontal normal crossings
Let $\pi:Y\longrightarrow \mathbf{P}^1_{\mathbf{Z}}$ be a finite surjective flat morphism of schemes, where $Y$ is a normal integral flat projective 2-dimensional $\mathbf{Z}$-scheme, with branch ...
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634 views
Branch locus of the Galois closure of a Belyi morphism
A morphism of curves is said to be Galois if the corresponding extension of function fields is Galois.
Let $f:X\longrightarrow Y$ be a finite morphism of smooth projective connected curves over ...
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429 views
What is the different in the cyclotomic tower over a finite ramified extension of Qp?
If $K_n$ is the field $\mathbb{Q}_p(\mu_{p^n})$, then it's easy to see that the relative different $\mathcal{D}(K_n / K_{n-1})$ is $(p)$ for all $n \ge 2$.
What happens if I take an arbitrary, ...
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Maximal (non-abelian) extensions of number fields unramified everywhere
Hello!
Let $K$ be a number field. All abelian unramified extensions are contained in the Hilbert class field which is a finite extension 'maximal' with respect to this property. For general ...
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1answer
368 views
Choosing tau for elliptic curves over the rational numbers with prescribed ramification data
Let $r>2$ and let $b_1,b_2,\ldots,b_r$ be in $\mathbf{P}^1(\mathbf{Q})$. Let $B$ be the divisor $$B:= \sum [b_i].$$ We consider this data to be fixed. For $d>1$, we define ...
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What to call the following variant of tame ramification
Suppose that $R \subseteq S$ is a generically separable extension of 1-dimensional normal domains (you can assume that $R$ is local if you'd like) of equal-characteristic $p > 0$ (for simplicity, ...
5
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1answer
689 views
What does the Riemann-Hurwitz formula tell us on the Picard variety
Let $f:X\longrightarrow Y$ be a finite separable morphism of smooth projective integral curves over an algebraically closed field.
Then we have a linear equivalence of Weil divisors on $X$: $$ ...
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Tame ramification of (mild) curve singularities.
Suppose that $C$ and $D$ are curves of finite type over an algebraically closed field $k$ (we make some of these hypotheses for simplicity). We view these as pointed curves with singularities $c \in ...
