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3
votes
1answer
101 views

The ample cone of a surface with an algebraic $\mathbb C^*$ action

Let $X$ be a compact complex protective surface that admits a nontirvial algebraic $\mathbb C^*$-action. It seems to me, that the ample cone of $X$ is polyhedral with finite number of faces. I wonder ...
3
votes
1answer
209 views

Does a projective variety have only finitely many associated Hilbert polynomials?

Let $X$ be a projective variety over $\mathbb{C}$. If $L$ is an ample line bundle, then $h_L$ denotes the Hilbert polynomial. Is it true that, if $L$ and $L'$ are ample line bundles which are ...
6
votes
1answer
201 views

Does ampleness descend along finite maps?

First, let me emphasize that for $X$ a not-necessarily proper variety, we say that a line bundle $L$ on $X$ is ample, if for some positive integer $n$, $L^{\otimes n}$ arises as $j^*O(1)$ for some (...
8
votes
0answers
146 views

Very ample divisors on blow ups of the projective plane

Let $X$ be $\mathbb{P}^2$ blown up at $k$ points in general position. The Picard group of $X$ is just $\mathbb{Z}^{k+1}$ and one knows the intersection product explicitly. If $D$ is an ample divisor, ...
0
votes
0answers
72 views

$H$ very ample, $f$ finite, is there uniform $C=C(\mathrm{deg}(f))$ for $C f^* H$ very ample?

Let $X$ and $Y$ be normal projective varieties over $\mathbb{C}$ of dimension $n$. Let $f: X \rightarrow Y$ be a finite morphism. Also, let $H$ be a very ample divisor on $Y$. Is there a constant $C=...
1
vote
1answer
129 views

Fiber product of projective varieties and ample line bundles

Let $X, Y$ be smooth, projective varieties, $L_X$ and $L_Y$ are very ample line bundles on $X$ and $Y$, respectively. If I understand correctly, $\mbox{pr}_1^*L_X \otimes \mbox{pr}_2^*L_Y$ is a very ...
2
votes
0answers
101 views

Do integral curves on simple abelian surfaces define big line bundles?

Let $A$ be a simple abelian surface over $\mathbb{C}$. Let $C\subset A$ be an irreducible and reduced one-dimensional closed subscheme. Since $A$ is simple, the normalization of $C$ is of genus at ...
5
votes
0answers
126 views

Is the stack of smooth canonically polarized surfaces uniformisable (or a good orbifold)

This is a question about the fundamental group of the moduli of canonically polarized surfaces. Let $\mathcal{M}$ be the (locally finite type separated Deligne-Mumford) algebraic stack of smooth ...
9
votes
0answers
323 views

Hartshorne's Conjectures about Algebraic Bundles?

In 1978 Hartshorne published a list of 26 open problems about algebraic bundles on projective spaces [Hart], proceeding from an Oxford conference organized by Atiyah. I understand that many of these ...
3
votes
0answers
147 views

When is the strict transform of very ample divisor ample?

Let $X$ be a projective variety and $Y \subset X$ is a very ample divisor on $X$. Let $Z \subset X$ be a regular subvariety (but $X$ need not be regular along $Z$) of codimension $2$ and $\pi:\tilde{X}...
1
vote
0answers
95 views

Does being big for a line bundle satisfy fpqc descent

Let $k$ be a field of characteristic zero, and let $L/k$ be a field extension. [Assume $k$ and $L$ are algebraically closed if necessary.] Let $X$ be a variety over $k$ and let $\mathcal{L}$ be a ...
1
vote
0answers
169 views

What is the support of a coherent sheaf on $X\times Y$ if it is invariant by tensoring a very ample line bundle on $X$?

Let $X$ be a smooth projective variety over a field $k$ with char$k=0$ and $\mathcal{L}$ be a very ample line bundle on $X$. Let $\mathcal{F}$ be a coherent sheaf on $X$. It is well-know that if $\...
5
votes
0answers
186 views

Extension of ample vector bundles is ample

As I read Huybrechts-Lehn's book on Moduli of Sheaves, it is making a claim that extensions of several (at least 2) ample vector bundles (on curves) is again ample. Somehow, I am unable to see this ...
2
votes
1answer
181 views

Pencils in very ample linear systems without curve in its base locus

If $L$ is a very ample line bundle over a smooth complex projective surface $X$ and $s_0, \dots, s_n$ is a basis of the global sections of $L$, is there some choice of $i,j$ such that the pencil ...
17
votes
3answers
1k views

Varieties with an ample vector bundle mapping to their tangent bundle

A well-known result of Andreatta and Wisniewski says: Let $X$ be a projective complex manifold whose tangent bundle $T_X$ contains an ample sub-bundle $\mathscr{E}$. Then $X$ is isomorphic to ...
2
votes
4answers
784 views

Global section of very ample line bundles and its value on stalks

Let $X$ be a projective scheme and $\mathcal{L}$ be a very ample line bundle on $X$ with respect to some projective embedding $X \hookrightarrow \mathbb{P}^n_{\mathbb{C}}$ (for some $n$). Given any ...
10
votes
2answers
691 views

On a proposition in Hartshorne's paper “Ample vector bundles on curves”

In Prop. 4.1, p. 87 of the article "Ample vector bundles on curves" (Nagoya Math. J. 43 [1971], 73--89), R. Hartshorne states the following: Let $A$ be an abelian variety [over an alg. closed field $...
1
vote
0answers
180 views

Uniqueness of lifting of very ample line bundle on smooth proper surfaces over DVR

Let $R$ be a complete Henselian discrete valuation ring of characteristic 0, $X_R$ be a surface smooth, proper and flat over $R$. Assume that the residue field $k$ of $R$ is algebraically closed of ...
1
vote
1answer
169 views

Big divisors and small transformations

Let $X$ be a smooth projective variety such that $-K_X$ is ample. Let $f:X\dashrightarrow Y$ be a small $\mathbb{Q}$-factorial transformation. I would like to know if is true or not that: $-K_Y$ is ...
1
vote
1answer
189 views

Ample divisors on $\mathbb{P}^n$ blown-up at $k$ general points

Let $X$ be the blow-up of $\mathbb{P}^n$ at $k$ general points. We can assume $k\leq n+4$. Let $$D = aH-b_1E_1-...-b_kE_k$$ be a divisor on $X$. Are there conditions on $a,b_1,...,b_k$ ensuring that $...
3
votes
0answers
154 views

Ampleness of Hodge bundles over complex curves

Let $C$ be a smooth, proper and connected curve over the complex numbers $\bf C$. Let ${\cal G}\to C$ be a smooth group scheme over $C$ and let $\epsilon_{\cal G}:C\to{\cal G}$ be its zero-section. ...
3
votes
1answer
360 views

Does normalization of projective varieties preserve very ampleness

Let $f:\tilde{X} \to X$ be a normalization of projective variety. Let $L$ be a very ample line bundle on $X$. Is $f^*L$ a very ample line bundle on $\tilde{X}$? If not true in general, is there any ...
4
votes
3answers
1k views

Weak Fano and Log fano varieties

A projective smooth variety $X$ is weak Fano if $-K_X$ is nef and big. We say that $X$ is log Fano is there exists a divisor $D$ such that $-(K_X+D)$ is ample and $(X,D)$ is Kawamata log terminal. Is ...
1
vote
2answers
292 views

An ample line bundle on a K3 surface

Let $X$ be a K3 surface obtained as a double covering of $\mathbb{P}^1 \times \mathbb{P}^1$ branching along a $(4,4)$-divisor. I think the natural line bundle $\pi^*\mathcal{O}_{\mathbb{P}^1\times \...
1
vote
1answer
1k views

Kleiman's and Nakai-Moishezon's ampleness criteria

I would like to work out a simple example to understand the relation between Kleiman ampleness criterion and Nakai-Moishezon ampleness criterion. Namely, let $X$ be the blow-up of $\mathbb{P}^{2}$ at ...
1
vote
1answer
199 views

A question on very ample line bundle on smooth projective surfaces

I had been reading a couple of texts by J.P. Demailly, one of them titled "Effective bounds for very ample line bundles". In the introduction the author mentions a result due to I. Reider (stated in "...
1
vote
1answer
319 views

Canonical bundle of the moduli space of curves

By the pointed canonical bundle formula the canonical bundle of $\overline{M}_{g,n}$ is given by $$K_{\overline{M}_{g,n}} = 13\lambda+\psi-2\delta-\sum_{I}\delta_{1,I}$$ where $\lambda$ is the Hodge ...
2
votes
0answers
195 views

ampleness in families

Let $X\to S$ be a smooth projective morphism with geometrically connected fibres over an integral noetherian regular scheme $S$ with generic point $\eta$. Let $L$ be a line bundle on $X$, and suppose ...
4
votes
1answer
219 views

Extension of linear system

Let $X$ be a normal irreducible closed subvariety of $\mathbb{P}^n$ and let $\Lambda$ be a linear system on $X$, without fixed component. Then, there exists a linear system $\Lambda'$ on $\mathbb{P}^n$...
1
vote
1answer
366 views

Does every ample divisor “span” a hyperplane?

Let $X\subset\mathbb{P}^n$ be a smooth projective variety of dimension $\geq 2$ and assume that it is not contained in any hyperplane. Now, take some hyperplane $H\subset\mathbb{P}^n$ and consider the ...
6
votes
0answers
370 views

(Relative) ampleness on algebraic spaces

This is a follow-up (of sorts) to this question. Let $f : X \to T$ be a proper morphism of schemes. Then the notion of a relative ample (or $f$-ample) line bundle can be defined in several ...
3
votes
0answers
164 views

Ampleness on the P^1 bundle over Siegel threefold

I am looking at the Shimura variety for $\mathrm{GSp}_4(\mathbb Q)$, with hyperspecial level structure at $p$. Let $X$ denote the special fiber over $\mathbb F_p$. For simplicity, let us pretend ...
1
vote
1answer
901 views

When is an ample line bundle on an abelian variety base point free?

So, any line bundle $L$ on an abelian variety $X$ determines a type $(d_1,\ldots,d_g)$ where $d_i|d_{i+1}$. It's well known that if $d_1\geq 3$ then $L$ defines an embedding, that if $L$ has no fixed ...
4
votes
2answers
666 views

When is the determinant of the push-forward of an ample line bundle ample

Let $f:X\to S$ be a "nice" morphism of "nice" schemes. Let $L$ be an ample line bundle on $X$. When is $\det f_\ast L$ also ample? A "nice" morphism could be anything from "finite type separated" to ...
0
votes
0answers
395 views

About first Chern class and Poincaré duality in case of an ample divisor

Led $D$ be a very ample divisor in $X$ projective variety. I can't understand why the first Chern class $c_1(\mathscr{O}_X(D))$ equals the Poincaré dual of $D$, $\mathscr{P}(D)$
0
votes
1answer
270 views

Sommese's theorem (generalized Weak Lefschetz) in arbitrary characteristic?

Sommese's theorem is a natural generalization of the Weak Lefschetz; for a smooth projective (connected) $X$, an ample vector bundle $E/X$ of rank $e$, and a section $s:X\to E$ it states that the ...
2
votes
0answers
461 views

The cohomology of the relative dualizing sheaf of a relative curve

Let $X\to S$ be a curve over $S=\mathrm{Spec} \ \mathbf{Z}$. Let $\omega$ be the relative dualizing sheaf of $X\to S$. Let $g$ be the genus of the generic fibre. Assume that $g\geq 2$. I know that $\...
4
votes
1answer
1k views

What can be said about a pullback of a very ample line bundle w.r.t birational maps?

Let $X$ be a smooth projective variety and $\phi: X \to \mathbb P^n$ be a map. If $\phi$ is an embedding then $E=\phi^*(O(1))$ is very ample. But can one say something if $\phi$ is birational (but not ...