Questions tagged [algebraic-curves]
for questions on one dimensional algebraic varieties over any field, including questions of moduli, and questions about specific curves.
986
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Question $B_5 \equiv B_1$ or $B_5 \ne B_1$?
Let $(C_1)$, $(C_2)$ be two conics on the same Ellipsoid, (or Hyperboloid, or Paraboloid). Let $A_1, A_2, A_3, A_4$ be four arbitrary points lie on $(C_1)$; $B_1$ be arbitrary point on $(C_2)$. The ...
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1
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Motivations to study the cohomology of the moduli space of curves
Could anyone give some interesting motivations to understand the cohomology of $\mathcal{M}_g$?
What I know: I have read the various approaches to construct $\mathcal{M}_g$ via orbit spaces for group ...
- 5,910
5
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0
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329
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Which equation of a Butterfly?
Let $A, B$ be two points and $L$ be a line on the Euclidean Plane. Take two points $J, G$ on the line $L$ such that $JG=constant$. Let $AJ$ meet $BG$ at $P$, $AG$ meet $BJ$ at $Q$, then the locus of ...
- 477
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310
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Max Noether's theorem for algebraic surfaces
The well-known Max Noether's theorem for curves [see Arbarello-Cornalba-Griffiths-Harris Geometry of Algebraic Curves vol.1, p. 117] states that, if a smooth curve $C$ is non-hyperelliptic, then the ...
- 65.1k
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233
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On tangent space to the fundamental group scheme
Let $X$ be a smooth, projective complex curve of genus at least $2$. If I understand correctly, after choosing a base point, one can associate to $X$, a fundamental group scheme $\pi$. I am trying to ...
- 2,126
14
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2
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3k
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Picard group of a singular projective curve
Let $X$ be a singular irreducible projective curve over an algebraically closed field and $\pi : \widetilde{X} \to X$ the normalization morphism. In the book on Neron models by Bosch et al. (I have ...
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240
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Why does the Galois twist of this cover specialize to a certain field extension?
I didn't feel MO was the best place to ask this question, so apologies for this, but when I asked it at https://math.stackexchange.com/questions/2297837/why-is-this-cubic-polynomial-generic-for-cyclic-...
- 1,102
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0
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112
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How to describe the subspace of invariants under the Rosati involution?
Consider the Jacobian $J_C$ of hyperelliptic curve
$$C\!: y^2 = x^5 + a$$
over a finite field $\mathbb{F}_p$, where $a \in \mathbb{F}_p^*$, $p \equiv 2 \ (\mathrm{mod} \ 5)$, $p > 2$. Let $\pi \...
- 1,801
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0
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594
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Automorphisms of semistable $G$-bundles
Let $X$ be a projective smooth curve over an algebraically closed field $\mathbb k$ of any characteristic. Let also $G$ be a reductive group over $\mathbb k$. (Probably) following Ramanathan, a $G$-...
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How much information is encoded in the Jacobian-Kummer K3 surface of a curve of genus two?
Assume we work over $\mathbb{C}$.
Let $S\subset \mathbb{P}^3$ be a quartic surfaces with 16 nodes (ordinary double points). Then there is a simple principally polarized abelian surface $(A,\theta)$ ...
- 1,015
11
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1
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Which curves have stable Faltings height greater or equal to 1
Let $Y$ be a smooth projective connected curve of genus $g>0$ over $\overline{\mathbf{Q}}$. Let $h_{\textrm{Fal}}(Y)$ be the Faltings height of $Y$.
Question 1. Can one classify or describe the ...
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2
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492
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What is the mod l monodromy of a generic trigonal curve?
For a hyperelliptic curve H, the mod 2 monodromy is smaller than $GSp_{2g}(F_2)$ -- since the two torsion of the Jacobian H is generated by differences of Weierstrass point, the monodromy of a generic ...
- 1,308
3
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1
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228
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Etale coverings of non-projective curves
For a smooth projective curve $Y$ over an algebraically closed field $k$ of characteristic 0, it is known that there exists a one-to-one correspondence between finite \'{e}tale morphisms $f:X\to Y$ of ...
- 37.3k
1
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0
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134
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Constructing embedded families of curves with general moduli
Is there a known way to construct flat families of smooth curves $\mathcal{C}/\mathcal{B}$ which are fiberwise embedded in a family of projective varieties $\mathcal{X}/\mathcal{B}$ and which have ...
- 1,971
4
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1
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Do only finitely many bisecants of a canonical curve intersect two distinct codimension 2 spaces simultaneously?
Setup & question
Let $C \hookrightarrow \mathbb{P}^{g-1}$ be a general canonical curve of genus $g \ge 4$ and let $Y_1,Y_2 \subset \mathbb{P}^{g-1}$ be codimension 2 linear subspaces such that $...
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The geometric genus under a generically finite to one rational map [closed]
Is it true that if $X$ and $Y$ are two irreducible algebraic curves over $\mathbb{C}$ and $f:X\dashrightarrow Y$ is a rational map that is generically n:1 for some $n\in \mathbb{Z}^{+}$ then the ...
- 1,044
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0
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68
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Centralizer/Normalizer of global sections of vector bundles on curves
Let $X$ be a smooth, projective curve of genus at least $2$ over $\mathbb{C}$ and $E$ be a vector bundle on $X$ of rank at least $2$. Given any point $x \in X$, denote by $S_x$ the image of the ...
- 1,949
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1
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385
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Endomorphism of globally generated sheaves on curves
Let $X$ be a smooth, projective curve (over $\mathbb{C}$) of genus at least $2$ and $E$ be a globally generated sheaf on $X$. I am looking for conditions/examples such that there exists a closed point ...
- 27.3k
3
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1
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582
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When every ideal containing $J(R)$ is an intersection of maximal ideals
Let $R$ be a commutative ring with $1$ such that every ideal containing $J(R)$, the intersection of all maximal ideals, is an intersection of maximal ideals. Is there any characterization for such a ...
- 4,669
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160
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An explicit description of $A_X$
Let $X$ be a smooth complex projective curve of genus at least $2$ and $p\in X$. Define $$A_X=H^0(X-p,\mathcal O_X).$$
By choosing a local parameter $z$ near $p$ we can see $A_X\subset Frac(\hat{O}_{X,...
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A sub-variety of a Grassmannian
Is there a nice description of the variety $G(r,2r) \setminus \sqcup_{i+j=r}(G(i,r) \times G(j,r))$ in terms of blow ups or a sub-variety of a secant variety or any other natural construction to see ...
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1
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When will the supporting hyperplane of a convex set coincide with the tangent?
Due to the supporting hyperplane theorem, a convex set $C$ in a separable topological space has supporting hyperplance at each of its boundary points. The theorem only guarantees its existence, now I ...
- 27k
3
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0
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114
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Projective normality of residual pencils on a general curve
Let $C$ be a general curve, say of even genus $g=2s$. Then $C$ has finitely many pencils $|L|$ of degree $\deg L=[g+3]/2=s+1$. Choose one such. The residual series is of degree $\deg(K_C-L)=3s-3$.
...
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2
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Line bundles and Cyclic Covers of Curves
Let $X \to Y$ be a cyclic etale cover of smooth projective geometrically connected curves over some field $k$. Then the map is classified by an element of the cohomology group $H^1_{et}(Y_{k_s}, \mu_n)...
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2
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620
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Rationality of curve does not depend on base change
By a curve I mean an integral one-dimensional scheme of finite type over a spectrum of a field.
Let $C$ be a curve over an arbitrary field $k$. It's probably a very well known fact, that $C$ is ...
- 267
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1
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390
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Is the number of ramified coverings of given degree of a curve with prescribed branch divisor finite?
Let Y be a smooth projective curve over C and prescribe a branch divisor B on Y. I want to know if the number of coverings of Y of fixed degree and branched along B is finite. If so, why? Or, where is ...
- 10k
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Cycles in the Chow ring of the moduli of curves coming from Hurwitz theory
Are there interesting cycles (other then the famous ones such as: Harris and Mumford's gonality divisor, the Gieseker-Petri divisor [which can be realized as the branch locus of a forgetful map from ...
- 1,971
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0
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147
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An analogue of Brill-Noether for hypersurfaces?
Let $d,g,r$ be natural numbers such that $d \geq 1$, $g \geq 2$, $r \geq 3$. Denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme classifying subschemes of $\mathbb{P}^r$ with Hilbert polynomial $P(x) = ...
- 1,971
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228
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De Jonquières formula vs. Relative GW invariants
Background. Let $C$ be a smooth projective curve of genus $g$. Denote by $C^d$ its d-th symmetric product and by $G^r_d(C)$ its associated variety
of linear series of type $\mathfrak{g}^r_d$, i.e.
$$ ...
- 1,971
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1
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431
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Example of projective variety that do not contain algebraic curves of genus strictly greater to $1$
Does exist a smooth, complex, projective variety $X$ of dimension $d\geq2$ such that $X$ does not contain smooth, complex, projective curves of wichever genus $g\geq2$?
Answer by Bertie: No, it does ...
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126
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Minimum number of generators for a gloablly generated sheaf over a curve
Let $X$ be a smooth, projective curve over an algebraically closed field and $E$ be a globally generated locally free sheaf of rank $r$. Is it always possible to write $E$ as the quotient of $r+1$ ...
- 2,195
1
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1
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228
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Local parameters and etale coverings of of elliptic curves
I found some absurd observation which I could not fix by myself. For an elliptic curve $E$ over $\mathbb{Q}$, let $\overline{E}=E\otimes\overline{\mathbb{Q}}$. Every multiplication-by-$n$ map $\...
- 4,058
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0
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181
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Analytic-Local Germs of "General" Sections
Let $C$ be an algebraic curve over an algebraically closed field $k$ of characteristic $0$, and let $\mathcal{L}$ be a base-point-free line bundle on $C$. Furthermore, let $p \in C$ be a smooth point, ...
2
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1
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Stabilization of semistable curves in a concrete case
Some days ago I learned for the first time about the stabilization of a semistable curve (from Knudsen's article), but I am still quite confused. If $C/k$ is a semistable curve (i.e. we allow rational ...
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Is quotient by maximal destabilizing sheaf, torsion-free?
Let $k$ be an infinite field (not necessarily algebraically closed), $X$ a smooth, projective curve over $k$ and $F$ a pure, coherent sheaf on $X$. Let $F'$ be the maximal destabilizing sheaf of $F$. ...
- 2,195
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0
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140
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Cohomology of the affine singular and projective smooth model
In the literature, in many places the computations of Cohomology
of the affine singular and projective smooth models are done
interchangeably. Let us work with curves. For example, if $\pi:Y\to \...
- 637
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1
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253
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Non-abelian group from affine hermitian curve
I was playing with the Hermitian curve $y^q + y = x^{q+1}$ over the field $GF(q^2)$ and chanced upon the following (non Abelian) group law on the points of the affine curve:
$(a,b) * (c,d) = (a+c,b+d+...
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2
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1
answer
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Fiber of the Prym map in dim 2
This must be very classical, but I can't find a reference.
Is there an explicit description of the (generic?) fibers of the Prym map $\mathcal{R}_3 \to \mathcal{A}_2$?
By this I mean the map ...
- 65.1k
4
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1
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335
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"Generalized" clutching maps between moduli spaces of curves
Let $P=\{1,\dots,n\}$ and $S\subseteq P$. The map $$\nu:\overline{\mathcal{M}}_{i,S\cup\{q\}}\to \overline{\mathcal{M}}_{g,P},$$ which attaches to a curve in the domain a pointed genus $g-i$ curve $[D,...
- 55k
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2
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Rigorous version of heuristic argument for genus-degree formula?
A recent MO question about non-rigorous reasoning reminded me of something I've wondered about for some time.
The genus–degree formula says that genus $g$ of a nonsingular projective plane curve of ...
CommunityBot
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18
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3
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628
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Putting algebraic curves in $\mathbb{R}^3$
Let $X \subset \mathbb{C}^2$ be a smooth algebraic curve. Thinking of $\mathbb{C}^2$ as $\mathbb{R}^4$, is there a smooth map $\phi: \mathbb{C}^2 \to \mathbb{R}^3$ so that $\phi: X \to \mathbb{R}^3$ ...
CommunityBot
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2
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Modular forms from counting points on algebraic varieties over a finite field
Suppose we are given some polynomial with integer coefficients, which we regard as carving out an affine variety $E$, for example:
$$ 3x^2y - 12 x^3y^5 + 27y^9 - 2 = 0 \tag{$*$} $$
(We might ...
- 36.1k
3
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0
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351
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Universal moduli space of rank two vector bundles over a curve
Let $\mathcal M_C$ denote the moduli space of rank two vector bundles over a smooth proper curve $C$ over an algebraically closed field $k$ of characteristic zero. Let $k(t)$ denote the function ...
- 18.3k
2
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222
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Reference request for lemma of Castelnuovo
While reading a paper by Ciliberto and Lazarsfeld ("On the uniqueness of certain linear series on some classes of curves") I came across a lemma of Castelnuovo used in the paper. The lemma says that ...
- 21
2
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1
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277
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On the stable model
I know that this is not a research question, but I am stuck from quite a while and I don't know what else to do.
Given a semistable curve $C$ over $k$, it is asserted that we may construct its stable ...
- 4,091
3
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1
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238
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Geometric (or at least non-cohomological) proof of Lefschetz trace formula for curves
There is an isomorphism between (rational) correspondences on a curve $C/\mathbb{F}_p$ orthogonal to the "valence zero" ones (i.e. orthogonal under intersection pairing to $\{*\}\times C$ and $C\times ...
- 66
0
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1
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202
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Prime divisors on the Jacobian of a genus 2 curve over $\mathbb{F}_q$ under the $n$ map
Let $H$ be a hyperelliptic curve geometrically irreducible of genus 2 over $\mathbb{F}_q$ with a rational point $\infty$ given by the model $y^2=f(x)$, where $f$ is monic of degree 5.
Consider the ...
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253
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Intuition behind results in Mumford's “Lectures on curves on an algebraic surface”, II
NOTE: This is a followup to my question here.
These are some questions concerning Mumford's "Lectures on curves on an algebraic surface".
We concern ourselves with questions of the Picard variety $P$...
CommunityBot
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6
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1
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Intuition behind results in Mumford's "Lectures on curves on an algebraic surface", I
These are some questions concerning Mumford's "Lectures on curves on an algebraic surface".
We concern ourselves with questions of the Picard variety $P$, and its dimension, of a complete nonsingular ...
3
votes
1
answer
496
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Name for curve?
I am doing something with the curve given parametrically by
$y = (-ar+b) r$, $x = \sqrt{r^2-y^2}$
for $r\in \lbrack (b-1)/a,b/a\rbrack$. It is nice enough (and of low enough degree) that I suspect ...
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