# algebraic-curves's questions - English 1answer

646 algebraic-curves questions.

### 2 An explicit correspondence for reductions of modular curves $Y(N)$

Let $Y(N)$ be the modular curve associated with the principal congruence subgroup $\Gamma(N) \subset \mathrm{SL}(2, \mathbb{Z})$ of level $N \in \mathbb{N}$. It is well known that this curve has a ...

### 1 When does the Jacobian of a smooth curve contains an unique principal polarization

Let $X$ be a smooth, projective curve of genus at least $4$ and $X$ non-hyperelliptic. I am looking for additional conditions on $X$ such that the Jacobian $J(X)$ of $X$ contains an unique principal ...

### 2 Naive question on the Jacobian of a curve

Let $X$ be a smooth, projective curve of genus $g \ge 2$. We know that the Jacobian $J(X)$ of the curve is a principally polarized abelian variety. The principal polarization is induced by the ...

### 5 Degeneration of smooth curves and Picard-Lefschetz formula

Let $\pi:\mathcal{C} \to \Delta$ be a family of projective curves of genus $g \ge 2$ over the unit disc $\Delta$, smooth over the punctured disc $\Delta\backslash \{0\}$ and central fiber $\pi^{-1}(0)$...

### 4 Priority for lemniscate of Gerono?

The Lemniscate of Gerono is a special case of the Lissajous curves. The dates for the two mathematicians are fairly close: Gerono (1799-1891) and Lissajous (1822-1880). There seems to have been ...

### 2 Why are modular curves non-trivial covers of the $j$-line

This is a very soft question. Let $n\geq 1$ and let $Y(n)$ be the (open) modular curve associated to $\Gamma(n)\subset SL_2(\mathbb{Z})$. Interpreted correctly, $Y(n)\to Y(d)$ is finite etale, ...

### 4 On a presentation of $\bar{M}_{2,1}$

In this answer it is said that $\bar{M}_{2,1}\cong \bar{M}_{0,7}/S_6$. However, I cannot see this. Given a curve of genus $2$ and a marked point, quotienting by the involution surely gives a rational ...

### 9 Natural model for genus $6$ curves

By Brill-Noether theory, the generic genus $6$ curve is birational to a sextic plane curve in $\mathbb{P}^2$. I was wondering if there is a direct/natural construction of this birational map. In other ...

### 1 Twisting a line bundle with the zero section

Let $X$ be a smooth projective curve and $L$ be an invertible sheaf on $X$. Denote by $\mathbb{L}$ the line bundle associated to $L$, $\pi:\mathbb{L} \to X$ the natural morphism and $0_\pi$ the zero ...

### 1 Image of curve along a finite etale Galois map

Let $f:X\to Y$ be a finite etale Galois morphism of varieties over $\mathbb{C}$. Let $C$ be a smooth quasi-projective connected curve in $X$. Is $f(C)$ a smooth curve?

### 3 Deformation of “Hecke modification”

Let $X$ be a smooth curve over $\mathbb{C}$. I wish to compute the deformation of the following data $(E,F,x)$. $E$ and $F$ are locally free sheaves over $X$ and $x$ is a point on $X$. They satisfy: ...

### 3 Is the isogeny class 1109.a of abelian surfaces in the LMFDB complete?

The LMFDB lists the Jacobian of the genus-2 curve 1109.a.1109.1 (http://www.lmfdb.org/Genus2Curve/Q/1109/a/1109/1) as being isolated in its rational-isogeny class. However, the LMFDB does not purport ...

### 3 Descent of coherent sheaves on finite coverings

Let $X$ be a non-singular hyperelliptic curve (over $\mathbb{C}$) and $\pi:X \to \mathbb{P}^1$ be a $2:1$ covering. Let $\sigma:X \to X$ be the hyperelliptic involution and $E$ be a locally free sheaf ...

### 6 Descent via an explicit isogeny (genus 2)

This question is related to a previous question posted by me here answered by Prof. M. Stoll. 5-Descent or ($\sqrt{5}$-Descent?) on certain genus 2 Jacobians. Here I ask some technicalities of a ...

### 6 Rational normal curves and tangent lines

Let $C,\Gamma\subset\mathbb{P}^n$ be degree $n$ rational normal curves in $\mathbb{P}^n$, such that for any $p\in C$ the tangent line $T_pC$ of $C$ at $p$ is tangent to $\Gamma$ as well. This means ...

### 15 Do $\mathbb{A}^1-S$ and $\mathbb{A}^1-\{0,1\}$ have a finite etale cover in common?

We work over the field of complex numbers. (But remarks in characteristic $p$ are very welcome.) Let $S$ be a finite set of points in $\mathbb{A}^1$ containing $0$ and $1$. [Edit: Assume $S$ contains ...

### 5 The degree of the Gauss map of Theta divisor

Let $R$ be compact a Riemann surface of genus $g$ and $J (R)$ be its Jacobian. For a subvariety $X$ of $J(R)$ of dimension $d$, denote the set of non-singular points of $X$ by $X_{reg}$. Then the ...

### 5 Closed but not rational points of a real cubic

In Mumford's Red Book of Varieties and Schemes, page 102, he gave the example of the closed but not rational points (that is to say points having residue field the complex field and not the real field)...

### 3 Meaning of general hyperplane $H$ in $\mathbb{P}^n$

Maybe this question is not suitable for this platform, I already put that same question in math.stackexchange and I find only vague answers. I'm studying the book of Rick Miranda; Algebraic Curves ...

### 3 The set of points of the fiber product of two algebraic stacks and the fiber product of the sets of points of two algebraic stacks

Let $\overline {\mathcal{M}_{g_{i}, n_{i}}}, \ i \in \{1, 2\},$ be a moduli stack of pointed stable curves of type $(g_{i}, n_{i})$ over a finite field $\mathbb{F}_{p}$. For any algebraic stack \$\...