All Questions
16 questions
1
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1
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141
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Why is $2A_0(X)=0$ for a cubic threefold $X$ containing a line, over an arbitrary field $k$
I can't quite follow Proposition $2.1$ of "UNIVERSAL UNRAMIFIED COHOMOLOGY OF CUBIC FOURFOLDS CONTAINING A PLANE". I posted this on Math stackexchange but got no answer.
Let $X$ be a smooth ...
- 679
3
votes
1
answer
255
views
About the contractability
Let $X\subset \mathbb{P}^3$ be the surface defined by the equation $xy-zw=0$, and consider the curve $E \subset X$ defined by the equation $x=z=0$.
Question. Can $E$ be contracted to a point?
- 328
1
vote
1
answer
501
views
Very ample + effective = ample?
Sorry if this question is not appropriate for this site, but I haven't got an answer on stackexchange. It's well known that there are divisors (on a normal projective variety over the complex numbers) ...
- 143
2
votes
0
answers
145
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Is there a name for a normal, projective variety where every effective divisor is ample?
Is there a name for a normal, projective variety such that every effective divisor is ample? Examples of such varieties are projective space, weighted projective spaces, and simple Abelian varieties ...
- 912
4
votes
1
answer
134
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Isomorphism outside of negative curves against the canonical
Let $X$ be a smooth projective complex variety and let us suppose that the closure of the union of curves $C$ on $X$ that are non-positive against the canonical divisor is a closed subset $F\subsetneq ...
- 7,722
1
vote
1
answer
729
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Push-forward of divisors and intersections
Let $f:X\rightarrow Y$ be a surjective finite morphism of varieties, with $X$ normal and $Y$ smooth. Let $D\subset X$ be a divisor and $C\subset Y$ a curve. Does the equality
$$C\cdot f_{*}D = f^{*}C\...
user168627
7
votes
1
answer
945
views
Push-forward of nef divisors via finite morphisms
Let $f:X\rightarrow Y$ be a finite morphism between smooth projective varieties, and let $D$ be an effective nef but not ample divisor on $X$.
Consider the divisor $f_{*}D$ on $Y$. Is $f_{*}D$ nef ...
user58018
4
votes
2
answers
764
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Linear sections of Segre varieties and rational normal scrolls
In a projective space $\mathbb{P}^{k+2}$ consider two complementary subspaces $\mathbb{P}^1,\mathbb{P}^k$, and let $C\subset\mathbb{P}^k$ be a degree $k$ rational normal curve. Fixed an isomorphism $\...
user114666
4
votes
1
answer
572
views
Pushforward of curves
Let $Z$ be a subvariety of an irreducible projective variety $X$, and let $i:Z\rightarrow X$ be the inclusion.
Let $N_1(X),N_1(Z)$ be the $\mathbb{Q}$-vector spaces of curves in $X$ and $Z$ ...
user114666
4
votes
0
answers
347
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Complete intersections in projective spaces
Let $X$ be an arbitrary smooth projective variety over a field $k$.
Do there exist:
a smooth complete intersection $X'$ in a projective space.
a surjective morphism of $k$-varieties $X'\to X$ ?
user119470
2
votes
0
answers
219
views
Liftability of varieties, after fpqc base change
Let $X$ be a smooth projective variety over a finite field, with a closed immersion to some other smooth projective variety $S$, with $S$ liftable.
Suppose there exists an fpqc cover $S'\to S$, such ...
user124171
1
vote
1
answer
239
views
Endomorphism of Chow group induced by a birational map
Let $\phi:X\dashrightarrow Y$ be a birational map between smooth projective $k$-varieties ($k=\bar k$) and $\Gamma$ be the closure of the graph of $\phi$. In Fulton's intersection theory example 16.1....
- 155
0
votes
1
answer
342
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Intersections of divisors in blow-ups of $\mathbb{P}^n$
Let $p_1,p_2,p_3\in\mathbb{P}^n$ be three general points, $X$ the blow-up of $\mathbb{P}^n$ at $p_1,p_2,p_3$, then along the lines $\left\langle p_i,p_j\right\rangle$, and finally along the plane $\...
user91659
5
votes
1
answer
1k
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Example of cone of numerically effective curves which is not polyhedral
I think I have seen more than one reference in which the cone of numerically effective curves can be 'not polyhedral', i.e. with an infinite number of extremal rays
I cannot remember where I read ...
- 2,215
2
votes
2
answers
627
views
Numerically negative exceptional divisor on a surface.
Suppose $S$ is an algebraic surface (possibly projective) over an algebraically closed field $k$. Suppose $D_i$ are irreducible smooth curves (rational, if you want) with negative self-intersection ...
- 2,215
2
votes
1
answer
307
views
F-curves and divisors on $\overline{\mathcal{M}}_{g,n}$
It is conjectured that a divisor on $\overline{\mathcal{M}}_{g,n}$ would be ample if $D\cdot C >0$ for all F-curves $C\subset \overline{\mathcal{M}}_{g,n}$. Does the intersection degree with all F-...
- 3,779