All Questions
33 questions
20
votes
3
answers
3k
views
Vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$
I have a question about vector bundles on the algebraic surface $\mathbb{P}^1\times\mathbb{P}^1$. My motivation is the splitting theorem of Grothendieck, which says that every algebraic vector bundle ...
- 545
16
votes
1
answer
4k
views
what is the cyclic cover trick?
What do people mean by the "cyclic cover trick"? I have found this expression a couple of times with no complete explanation, both talking about curves and surfaces...
- 3,779
15
votes
6
answers
3k
views
Curves with negative self intersection in the product of two curves
I wonder if the following is known: Are there two compact curves C1 and C2 of genus>1 defined over complex numbers, such that their product contains infinite number of irreducible curves of negative ...
- 28.9k
14
votes
2
answers
3k
views
Motivation for birational geometry
I'm interested in how do people that work in birational geometry view their field — specifically, what are the kinds of geometric questions (as opposed to commutative-algebraic questions) that ...
- 251
9
votes
1
answer
1k
views
Finite morphisms to projective space
Let $X$ be a projective variety of dimension n. Then there exists a finite surjective morphism $X \to \mathbf P^n$. Let $d$ be the minimal degree of such a finite surjective morphism.
Let $d^\prime ...
- 91
6
votes
3
answers
1k
views
Are there (-2)-curves on an Enriques surface?
Let $X$ be an Enriques surface. A $(-2)$-curve is an irriducible rational curve on X such that $C^2 = -2$. By Proposition [VIII,16.1] from Barth-Peters-Van de Ven, we have that if $D^2 = -2$, then it ...
- 427
6
votes
1
answer
1k
views
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 ...
user97565
6
votes
2
answers
422
views
Nef divisors on surfaces
Let $X$ be a smooth projective rational surface over an algebraically closed field of characteristic zero, and $D$ a divisor on $X$ such that $D$ is nef and $D^2 = 0$ with the following properties:
$...
- 8,998
6
votes
0
answers
674
views
Bezout Theorem in $\mathbb P^3$
For a $\mathbb C P^2$ is known a result: if through the generic points $p_1,p_2,\dots p_n$ with multiplicities $m_1,m_2\dots, m_n$ correspondingly a degree $d$ curve passes then $d^2\geq m_1^2+m_2^2+\...
- 5,063
5
votes
3
answers
497
views
Is there a way to find any non-trivial $\mathbb{F}_p(t)$-point on the given elliptic curve?
Consider a finite field $\mathbb{F}_p$ (where $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$) and the elliptic curve
$$
E\!:y^2 = x^3 + (t^6 + 1)^2
$$
over the univariate ...
- 1,833
5
votes
1
answer
306
views
Some questions about the (projectivized cotangent bundle of the) symmetric square of a genus $3$ curve
Let $C$ be a smooth, non-hyperelliptic curve of genus $3$ and $X:= \mathrm{Sym}^2{C}$ its symmetric square. Then $X$ is a smooth, minimal surface of general type with $p_g=q=3, \, K^2=6$.
Calling $\...
- 66.3k
5
votes
1
answer
393
views
surface with rational curve in the double locus
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 \...
- 379
5
votes
0
answers
326
views
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 ...
- 66.3k
4
votes
1
answer
665
views
Polars of algebraic curves and surfaces
I asked this on Math.StackExchange, but received no response, so trying here ...
A paper I'm reading says the following ...
With homogeneous coordinates $\mathbf{x} = [x,y,z,w]$, let $F(\mathbf{x})...
- 649
4
votes
1
answer
199
views
Jacobians $\mathbb{F}_q$-isogenous to the direct square of an ordinary elliptic $\mathbb{F}_q$-curve of $j$-invariant $0$
Consider an ordinary elliptic curve $E_b\!: y^2 = x^3 + b$, of $j$-invariant $0$ over a finite field $\mathbb{F}_q$, such that $\sqrt{b} \not\in \mathbb{F}_q$.
Question. What are some examples of ...
- 1,833
4
votes
1
answer
674
views
Genus two pencil in K3 surface
It is known that smooth $K3$ surface can be obtained as two fold branched cover of rational elliptic surface $E(1) = \mathbb{CP}^2 9 \bar{{\mathbb{{CP}^2}}}$ along the smooth divisor $2F_{E(1)} = 6H - ...
- 87
3
votes
1
answer
414
views
Modern reference for the theory of correspondences for curves
The classic theory of correspondences between smooth algebraic curves can be found in André Weil's Foundations of algebraic geometry. However, this reference works in a pre-modern algebraic geometry ...
- 2,871
3
votes
0
answers
287
views
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 $(A,\theta)$ ...
- 1,025
3
votes
0
answers
255
views
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 $P$...
user97565
3
votes
0
answers
298
views
What does Hodge theory tell us about simply connected surfaces of general type
Let $X$ be a smooth complex projective variety. We know that $\Omega^1_X$ has a non-zero section if and only if the abelianization of the fundamental group of X is infinite. This follows from Hodge ...
- 43
2
votes
1
answer
328
views
Complete Linear system on Del Pezzo surfaces
Is there always a reducible curve (EDIT: with exactly two irreducible components intersecting in at least 2 points) in a complete linear system (EDIT: of dimension at least 2 with curves of genus at ...
- 453
2
votes
1
answer
386
views
Surfaces singular along a curve
Let $C\subset\mathbb{P}^3$ be a smooth curve a degree $d$ and genus $g$. Let $\mathcal{S}$ be the system of surfaces of degree $k$ in $\mathbb{P}^3$ containing $C$ with multiplicity $\beta$.
What is ...
user47036
2
votes
1
answer
182
views
Does $\omega_C\simeq N_{C/S}$ always happen on Enriques surfaces?
Let $S$ be an Enriques surface and $C\subset S$ a smooth irreducible curve of genus $g$.
Consider the condition $$\omega_C\simeq N_{C/S}$$
For example, when $g=1$ then $\omega_C=\mathcal{O}_C$ and ...
- 761
2
votes
1
answer
234
views
(Non-)Rationality of a certain quotient of the symmetric square of the Fermat sextic (quartic) curve
Consider the Fermat sextic curve $F: x^6 + y^6 + 1 = 0$ over an algebraically closed field of characteristic $0$. It has the two order $3$ automorphisms $\omega_x(x,y) := (\omega x, y)$ and $\omega_y(...
- 1,833
2
votes
1
answer
345
views
discriminant of smooth quartic del Pezzo surface in $\mathbb{P}^4$
I can't understand the proof of Lemma3.3 in Stability of genus 5 canonical curves.
Let $C$ be a complete intersection of three quadrics in $\mathbb{P}^4$ and let $\Lambda$ be the net of quadrics ...
- 69
2
votes
0
answers
65
views
What conditions are sufficient for two points to be independent in the Mordell-Weil group?
Consider a finite field $\mathbb{F}_{q}$ and an elliptic surface
$$
\mathcal{E}\!: y^2 + a_1(t)xy + a_3(t)y = x^3 + a_2(t)x^2 + a_4(t)x + a_6(t),
$$
where $a_i(t) \in \mathbb{F}_{q}[t]$. I am mainly ...
- 1,833
2
votes
0
answers
137
views
Is a supersingular Kummer surface $k$-unirational in characteristic 2?
Let $k$ be a perfect field of even characteristic. Consider the simplest example of a supersingular genus 2 curve, i.e.,
$$
C\!: y^2 + y = x^5.
$$
By the article of J. S. Müller "Explicit Kummer ...
- 1,833
2
votes
0
answers
289
views
quasi-trigonal curves
I have read in the literature about quasi-trigonal curves. Such a curve C is a hyperelliptic curve X with two points p,q identified (basically a pinch). They seem to be pretty important in the theory ...
- 3,779
1
vote
1
answer
261
views
Constructing a curve with good reduction over a function field
Let $K$ be the function field of a smooth projective connected curve $B$ over $\mathbf{C}$.
Let $g\geq 0$ be an integer.
Does there exist an nonsingular integral $\mathbf{C}$-scheme $X$ with a ...
- 13
1
vote
1
answer
2k
views
Intersection number of divisors with its pull back and its push forward
I am in an ideal situation but I would appreciate a hint. First here is the scenario.
Let $\mathcal{J}$ be an the abelian variety obtained from the Jacobian of a genus $2$ curve $\mathcal{H}/k$ ...
1
vote
0
answers
262
views
Section ring $R(X,D)$ of $D$ is finitely generated if $\kappa (X,D) \leq 1$
In the remark on the bottom of page 5 of this paper, the author states that
It is well-known fact that the algebra of a Divisor $D$ with $\kappa (X,D) \leq 1$ is finitely generated over $k$. In ...
- 61
0
votes
1
answer
323
views
Morphisms contracting a family of curves
Let $f:X\rightarrow Y$ be a morphism of normal projective varieties. Let $S\subseteq X$ be a surface admitting a morphism $g:S\rightarrow C$ to a curve $C$ such that any fiber of $g$ is a curve.
...
user58018
0
votes
0
answers
51
views
Does this tangent developable-like surface have a cusp along a curve or is it smooth?
Consider the following surface $X$ which is a subvariety of the full flag variety $Y=\{0 \subset V_1 \subset V_2 \subset V\}$ where $V$ is a fixed three-dimensional vector space and ${\rm dim} V_i =i$....
- 2,964