**653 algebraic-curves questions.**

If $X$ is a curve with a nodal singularity at $x$, it's referred to here and here that its dualising sheaf is
$$\omega_X \ = \ \pi_*(\Omega_{X}(p_1+\cdots+p_n)').$$
Here, $\pi:X\to X'$ is the ...

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 ...

I am interested in the existence of a surface $X$ over $\mathbb{C}$ with the following properties (or a reason for why one cannot exist):
$X$ is slc (and not-normal)
There is rational curve $C \...

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?

(I'll ask the question over $\mathbb{R}$, but feel free to change fields if that makes the answer more straightforward or more interesting.)
Let $Q$ denote a bivariate quadratic:
$$Q(x,y) = Ax^2 + ...

$\newcommand{\F}{\mathcal{F}}$
$\newcommand{\ox}{\mathcal{O}_X}$
Let $f:X \to \operatorname{Spec}(k)$ be a projective scheme of dimension one over a field $k$. The Riemann-Roch equation for such ...

Consider relative curves $X \to S$, defined to be flat, integral, projective schemes of relative dimension 1 over $S$. When are these objects determined by their fibers?
So if $X,Y$ are $S$-schemes ...

I am interested in considering examples of abelian varieties that are principally polarized with CM in dimension three. However, I am struggling to construct or find even a single instance.
In ...

I am trying to understand the difference between Cohen Macaulay and Locally Cohen Macaulay curves.
The stacks project https://stacks.math.columbia.edu/tag/02IN says that a scheme (a curve in ...

Let $X$ be a nodal curve, possibly reducible. Then can any torsion free sheaf of rank one on $X$ be expressed as $\pi_*(L)$, where $L$ is a line bundle on a partial normalization of $X$? This looks ...

In what publication the following theorem is proved?
Let $a$,$b$, $c$ be real numbers and we choose a Lyapunov function for the system:
$$
\left\{
\begin{array}{cr}
\dot{x} = y \\
\dot{y} = z \...

Given a curve in $\mathbb{C}^2$ that is parametrized by polynomials $ x(t), y(t) \in \mathbb{C}[t]$, it is possible that for distinct $t_1$ and $t_2$, we have an intersection or crossing $(x(t_1), y(...

Let $k$ be a field, let $X/k$ be a nodal curve (which means $X_{\overline{k}}$ is a connected reduced proper curve with at worst nodal singularities.)
Does there always exists a proper flat family $\...

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 ...

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 ...

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 $\pi^{-1}(0)$...

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 ...

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, ...

The Faltings theorem states that the number of rationals over an algebraic curve is finite if the genus is greater than 1. The genus decreases by increasing the number of singularities. My question is ...

Let $L$ be a line bundle on a smooth connected complete complex algebraic surface $X$. Assume that $L$ has enough sections i.e. that $H^0(L,X)$ has dimension $> 1$. A nonzero section $s$ of $L$ ...

I hope this question belongs here. The situation in this question is quite particular and specific.
I am trying to weak some of theory to measure the degree of some function on the Jacobian of a ...

Let $K$ be a field equipped with a non-Archimedean absolute value, let $\Gamma$ be a Schottky group in $PGL_2(K)$, and let $X_\Gamma$ be the associated Mumford curve, which is a proper smooth rigid ...

A year ago I found the Pascal theorem for three dimentions as follows:
Let $(C_1)$, $(C_2)$ be two conics on the same Ellipsoid, (or Hyperboloid, or Paraboloid). Let $A_1$, $A_2$, $A_3$, $A_4$, $A_5$,...

I am trying to understand examples of differing curves with the same Jacobian, and the quirks of the Jacobian.
Here is my question: What is an example where $Aut(Jac(X, a))$ and $Aut(Jac(X', a'))$ ...

I need to know what can be said in general about the tangent bundle of reducible curves over complex numbers with arithmetic genus one, say $I_N$.
As far as I know for any Simpson semistable torsion ...

For my thesis, I'm working on the intersection of projective plane curves over $\mathbb{C}$. We define the intersection number of projective plane curves (see for example Gibson - Elementary Geometry ...

There is a mistake in the following argument but I cannot see where. Can someone help me, please?
Let $C$ be any smooth curve of genus $g\geq 1$ and $D$ a general effective divisor of degree $g+2$. ...

Let $C$ be an hyperelliptic curve with an involution $\sigma$, and let $E$ be a rank $2$ stable involution invariant vector bundle on $C$, that is $\sigma^{*}E \cong E$. Let $(End(E) \otimes K_C)_{+}$ ...

The underlying field is $\mathbb{C}$.
Let $\pi:\mathcal{C} \to \mathbb{A}^n$ be a flat family of projective curves (not necessarily smooth) of genus $g \ge 2$. Assume $\mathcal{C}$ is regular. Let $\...

Over an algebraically closed field of characteristic zero, e.g. $\overline{\mathbb{Q}}$, the Deligne-Mumford stack $\overline{\mathcal{M}}_{g,n}$ represents the functor $$\overline{\mathcal{M}}_{g,n}(...

Consider an algebraically closed field $k$ of prime characteristic. It is widely known that up to $k$-isomorphism there is a finite number of supersingular elliptic curves over $k$. Let $C$ be a ...

Let $\mathcal{M}_2$ be the moduli space of genus two curves and $\mathcal{A}_2$ the moduli space of principally polarized abelian surfaces. Then the Abel-Jacobi map gives an open embedding $\mathcal{M}...

I would like to work through computing the blow-up of a particular curve along a subvariety consisting of just two points, both of which are ordinary double points.
Let $$F(X,Y,Z) = X^4 + Y^4 - X Y^2 ...

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 ...

In his 92 paper, Mukai showed that a general genus $6$ curve may be represented in $\mathbb{P}^9$ as the intersection of the Grassmannian $G(2,5)$ (under the Plucker embedding), a plane $H\cong \...

Let $C$ be a smooth projective curve over $\mathbb{C}$. We will be working in the topological category. Let $A \in D^b(C):=$ bounded derived category of sheaves on $C$. Is it true that $$A \cong \...

Let $M_{g, n}$ be the moduli space of curves of genus $g$ with $n$ marked points. Let $M_{g, \vec{n}}$ be the moduli space of marked curves with a choice of a (possibly zero) tangent vector at each ...

Let $C$ be an hyperelliptic curve of genus $g$ and let $f:C\rightarrow
\mathbb{P}^1$ be the corresponding 2 to 1 covering
ramified in $2g+2$ points.
Let $L$ be a line bundle on $C$ such that either $...

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 ...

Let $\varphi\!: S \to S^\prime$ be a birational morphism of projective non-singular irreducible surfaces over a (algebraically closed) field $k$ and let $D \in \mathrm{Div}(S)$. Also, let $\big(\...

Let $C$ be a smooth curve of genus $g$ over $\mathbb{C}$. I am interested in the following property:
There exists a point $P \in C$ such that $K_C \sim_\mathbb{Q} (2g-2)P$. Equivalently, $K_C - (2g-2)...

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 ...

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:
...

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 ...

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 ...

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 ...

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 ...

Let $C$ be a smooth projective curve over an algebraically closed field $k$, of genus $g$.
It is well known that, after fixing a point $p_0$, the map $C^{(n)}\to J$ sending $\{a_1,\dots,a_n\}$ to $[...

I would like to suggest me some good books on (Complex) Algebraic Curves.
I am searching a book for Undergraduate-Begginer Level in this part of mathematics, the algebraic curves. I found some books ...

Let $C$ be a $1$-dimensional proper finite type scheme over field $k$, such that $p_a(C)=0$ (or equivalently $h^0(\mathcal{O}_C)-h^1(\mathcal{O}_C)=1$), then is it true that the reduced curve $C_{red}$...

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