# Tagged Questions

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### How to compute this Riemann surface?

This question is related to other more general question that I asked Computing Riemann surfaces of a given algebraic function. By the way, I've found an approaching in Markushevich's book that ...
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### Example of a Regular Map

I am working with Shafarevich's "Basic Algebraic Geometry 1". Example 1.15: The map $f(t)=(t^2,t^3)$ is a regular map on the line $\mathbb{A}^1$ to the curve given by $y^2=x^3$. I am not ...
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### References for elliptic curves over schemes

As in the title, I want some references about theories for elliptic curves over rings(not fields) or over schemes. I heard that behaviours(?) of such elliptic curves are not as simple as elliptic ...
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### Does $\,f_* \mathcal{O}_{X_T} \cong \mathcal{O}_{T}$ hold in this situation?

Let $X$ be a scheme over $S$ and consider the following hypothesis : \begin{cases} \; (1) \quad f:X\to S \text{ is quasi-compact and quasi-separated } \\\\ \; (2) \quad f:X\to S \text{ admits a ...
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### Tracing down (or pushing forward) differentials and Abel Jacobi map on principal divisors

Let $X$ be a complete one-dimensional curve over $\Bbb C$ (please add any hypotheses in case I forgot them). Then we have the Abel-Jacobi map $$\mathcal{AJ}: \text{Div}_0 \to \Bbb C^g/ \Lambda$$ ...
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### Local parameter of curves in affine n-space

I'm looking for a double answer to this question: a mathematical one (say, if the statement is correct or not) and a philosophical one (say, why we do expect this to be true, or not). Let $k$ be a ...
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### Curve with acnodes over closed fields?

From Wikipedia: An acnode is an isolated point not on a curve, but whose coordinates satisfy the equation of the curve. The term "isolated point" or "hermit point" is an equivalent term. I was ...
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### Computing Riemann surfaces of a given algebraic function

I've never seen written in a book a way or an algorithm for computing Riemann surfaces of a given algebraic function. I would like to know how to construct such Riemann surface using intuitive cutting ...
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### Exercise 1.11 of Eisenbud

I'm doing the exercises from Eisenbud's Commutative Algebra with a view toward Algebraic Geometry, and I don't understand part of one of them, ex. 1.11 a): Exercise 1.11 a: Over $\mathbb{C}$, ...
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### Simple proof of an equality on curves, with or without local fields

Is there a simple way to see that, given a nonsingular curve $X$ and a finite morphism $f\colon X\to Y,$ then $$2\cdot c_1(f_*\mathcal O_X)=-f_*R_f,$$ where $R_f$ is the ramification divisor of $f?$ I ...
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### Pullback of an ample line bundle through a projection

Assume $X_1$ and $X_2$ are two smooth projective curves and let $M_i$ be an ample line bundle on $X_i$, for $i=1,2$. Further, denote the natural projection map on the $i$-th factor by ...
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### Geometric interpretation of the Riemann-Roch for curves

Let $X$ be a smooth projective curve of genus $g\geq2$ over an algebraically closed field $k$ and denote by $K$ a canonical divisor. I have some clues about the geometrical interpretation of the ...
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### Given $\omega_i \in \Omega_X(U_i)$ can I find $f\in {\cal O}_X(\cap U_i)$ so that $df = \omega_1 - \omega_2$

As per this question: Duality in algebraic de Rham cohomology I am trying to show that the map $H^1(X,\Omega_X) \rightarrow H^2_{\text dR}(X/k)$, where $X$ is a projective algebraic curve over an ...
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### What is this cycle on the Jacobian of a curve?

Let $C$ be a curve and $J$ it's Jacobian. There is the standard Abel-Jacobi map $a:C\rightarrow J$ which is given by $Q\rightarrow Q-P$ for some fixed point $P$ (here I am regarding $J$ as the degree ...
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### What is normal crossing?

I could not find any reference for normal crossings. The definition here is not so clear to me. In some texts, they sometimes said that two varieties have normal-crossing (non-normal crossing) with ...
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### Vanishing of $R^1f_*\mathcal O_X$

I am probably missing something obvious here, but none the less, here goes: Is the following statement (or perhaps some minor modification of it) true and if so, why: $R^1f_*\mathcal O_X = 0$ for a ...
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### Smoothness of the Picard group of a smooth curve

Let $X$ be a smooth projective curve over $k=\bar{k}$ and denote its Picard group by $\operatorname{Pic}(X)$, with the usual scheme structure coming from the representability of the relative Picard ...
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### Prove that a curve in P^n of degree n not contained in a hyperplane is rational

The set up is as stated above. We have a projective curve $X$ of degree n embedded in $\mathbb{P^n}$, which is not contained in any hyperplane. We claim that it is therefore rational. The way I have ...
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### Duality in algebraic de Rham cohomology

I am trying to prove that the following is a short exact sequence $$0 \rightarrow H^0(X,\Omega_X) \rightarrow H^1_{\text {dR}}(X/k) \rightarrow H^1(X,\mathcal O_X) \rightarrow 0,$$ where $k$ is an ...
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### Vector bundle on a transversal intersection

Consider a curve $X$ (over a field $k$), which is an union of two smooth and connected curves $C$ and $D$ meeting at exactly one point $P \in X$ transversally. Let $E$ be a vector bundle on $X$ such ...
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### existence of a line on a cubic surface

I'm trying to understand the proof in Miles Reid's book: Undergraduate Algebraic Geometry, that: "there exists at least one line l on S", where S is a non-singular cubic surface. In the book the proof ...
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### Singular varieties

Let $y^2=x^3+ax+b$ and V be its affine variety. V is singular iff $y^2-x^3-ax-b$, 2y, and $3x^2+a$ have a common zero iff $x^3+ax+b$ and $3x^2+a$ have a common zero iff $x^3+ax+b$ has a multiple root. ...
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### no morphism between curves of different genus

Let $C_1,C_2$ be two smooth irreducible curves of genus 4,3 resp. Prove there is no morphism $\phi: C_1 \to C_2$. Well, the tool of treating genera is Hurwitz's theorem, which says here that ...
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### Selecting a book for a reading course in Algebraic Geometry

I'm not sure if this is an appropriate question in this forum, but here is the situation. I must begin by saying that I know basically nothing about Algebraic Geometry, but this semester I will be ...
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### Finding all morphism from a variety to itself

Let $$C:=X^2+Y^2-Z^2$$ be a projective variety in $\Bbb P^2$. What are all the morphisms $C\to C$ ? More generally, how does one find all morphisms from a given variety to itself? ...
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### $Im(\phi)$ is closed subset of $\mathbb{A}^2$

let $\alpha(t)$ and $\beta(t)$ $\in$ $K[t]$ , $\phi(t)=(\alpha(t),\beta(t))$ is a morphism from $\mathbb{A}^1$ to $\mathbb{A}^2$ show that $Im(\phi)$ is closed subset of $\mathbb{A}^2$. it seems ...
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### Normalization of a curve

This is a point in an exercise given during my Commutative Algebra course. Let $k$ be an algebraically closed field with characteristic different from $2$. Let $R=\frac{k[x,y]}{(y^{2}-f(x))}$, where ...
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### Threefold category equivalence: algebraic curves, Riemann surfaces and fields of transcendence degree 1

According to this article, The category of curves over the complex numbers is equivalent to two other categories: Riemann surfaces and ﬁelds of transcendence degree 1 over $\mathbb{C}$. But I ...
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### Two conics have exactly one intersection point

We have two conics $Q_1,Q_2$ on $\mathbb{P}_2$ over some algebraically closed field. Also $Q_1$ and $Q_2$ are supposed to be smooth. I've just discovered Bezout's theorem, which states that two ...
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### computing the divisor of a differential

I have some trouble to computer the divisor of a differential in the subject of algebraic curve. Any feedback is greatly appreciated.Thank you.
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### $V(f)$ is irreducible iff $f=g^k$, $g$ irreducible

I'm trying to prove this theorem $V(f)$ is irreducible iff $f=g^k$, $g$ irreducible. To prove the converse, we have $V(f)=V(g^k)=V(g)$, since $g$ is irreducible $V(g)$ is irreducible, then ...
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### $H^1$ of a constant sheaf

Let $X$ be an irreducible smooth curve, and $\underline{k(X)}$ the constant sheaf on $X$ with the function field $k(X)$ as fibers. Reading from Serre's Algebraic groups and class fields I met the ...
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### Linear Equivalence of Divisors on Projective Plane Cubic

I'm self-studying Miranda's Algebraic Curves and Riemann Surfaces and am uncertain of how I'm supposed to solve problem V.2c on linearly equivalent divisors: Let $X$ be the projective plane cubic ...
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### Finding a Uniformizer of a Discrete Valuation Ring

Suppose I have a discrete valuation ring. Then what are some techniques for explicitly finding a uniformizer? I'm especially interested in situations where the ring is given similarly to the following ...
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### For this morphism of integral schemes $X\to Y$, if $X$ is geometrically reduced (irreducible) then is $Y$ also geometrically reduced (irreducible)?

Let $k$ be an arbitrary field. Suppose that $C/k$ is an integral curve which is birationally equivalent to a projective line. Is it true that $C$ is geometrically reduced and irreducible? This is of ...
Suppose that $C$ is a normal projective curve over some base field $k$, possibly of positive characteristic. I am wondering to what extent one can modify the base field $k$ while not braking the ...
Let $X\overset{f}\to S$ be a flat morphism of schemes, and $S=\operatorname{Spec}(k)$ for an algebraically closed field $k=\bar{k}$. I'm just interested, for the moment, with schemes $X$ corresponding ...