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0
votes
1answer
139 views

A covering lemma of Kawamata

In the paper "A generalization of Kodaira-Ramanujam's vanishing theorem", Kawamata states a covering lemma (Lemma 5) which is Let $X$ be a non-singular projective variety, and $D$ be a divisor ...
7
votes
1answer
234 views

Top self-intersection of exceptional divisors

Let $Y\subset\mathbb{P}^n$ be a smooth variety of codimension two. Consider the blow-up $X = Bl_Y\mathbb{P}^n$ of $\mathbb{P}^n$ along $Y$, and let $E$ be the exceptional divisor over $Y$. Then $E$ ...
6
votes
3answers
269 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
169 views

When is a blow-up of $\mathbb{P}^2$ a Mori Dream Space?

Let $p_1,...,p_k\in\mathbb{P}^2$ be general points. Let us consider the blow-up $X_k = Bl_{p_1,...,p_k}\mathbb{P}^2$. It is clear that if $k\leq 3$ then $X_k$ is toric and hence a Mori Dream Space. ...
0
votes
0answers
60 views

Compactification of the affine space with a del Pezzo surface

Is there some nice compactification of the affine space with a del Pezzo surface of degree $\le 4$ (for example with a cubic surface) ? More precisely, I would like a projective algebraic variety $X$ ...
2
votes
1answer
141 views

Recognizing a Mukai flop

Let us first review the usual construction of a Mukai flop. Suppose $M$ is a smooth $2m$-dimensional projective variety over $\mathbb C$ containing a closed $m$-dimensional subvariety $W$, and suppose ...
2
votes
1answer
113 views

Properties of extreme rays

Let $X$ be a projective variety over $\mathbb{C}$, then the effective curves module the numerical equivalence form a cone. For my understanding, if the extreme ray $[C]$ has the property that ...
1
vote
1answer
121 views

Kawamata-Log-Terminal pairs

Let $p_1,...,p_n\in\mathbb{P}^3$ be general points, and let $\Delta\subset\mathbb{P}^3$ be a general surface of degree $d$ with points of multiplicity $m_i$ at $p_i$ for $i = 1,...,n$. Consider the ...
1
vote
1answer
119 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 ...
3
votes
1answer
135 views

$\mathbb{P}^1$-fibrations over $\mathbb{P}^2$ that are not rational

We know that if $X$ is a smooth complex projective variety and we assume that there is a dominant morphism $f : X \to Y$ with $Y$ and the general fibers of $f$ rationally connected. Then $X$ itself is ...
1
vote
0answers
58 views

How to Calculate Minimal Log Discrepancy on a Toric Variety?

Let $\sigma$ be a $3$ dimensional strongly convex rational polyhedral cone on $\mathbb{R}^3$ and $X_\sigma$, the corresponding affine toric $3$-fol. Also, assume that $\Delta$ is an anti-boundary ...
1
vote
0answers
74 views

Can discrepancy change after pseudo isomorphisms?

Let $X$ and $Y$ be two projective irreductive algebraic varieties of dimension $3$ and let $f:X\dashrightarrow Y$ be a pseudo isomorphism, i.e. a birational map which restricts to an isomorphism ...
3
votes
2answers
111 views

Irreducible divisor in a basepoint free linear system

Let $X$ be a projective, normal variety over complex field with canonical singularities. Suppose $|D|$ is a basepoint free linear system, then is it true that the generic elements in $|D|$ are ...
3
votes
1answer
117 views

Blowing up rational singularities

Let $X$ be a projective surface embedded into $\mathbb{P}^n_{\mathbb{C}}$ having at most rational singularities. Let $\tilde{X} \to X$ be the minimal resolution of $X$. Is it possible to embed ...
8
votes
1answer
433 views

Why is the standard flop a flop?

I have seen at least two ways to define flops (and similarly flips). We start with $Y \to X$, a surjective birational morphism, contracting a locus of codimension at least 2, such that $K_Y$ is ...
10
votes
1answer
578 views

is the Hodge conjecture birationally invariant?

Let $X$ and $Y$ be two birational smooth projective varieties over the complex numbers. Assume $X$ satisfies the Hodge conjecture. Is it known that the Hodge conjecture holds for $Y$?
0
votes
1answer
70 views

Resolution of the indeterminacy locus of a rational map away from an irreducible component

Suppose I have a rational map between two smooth, complex, projective varieties (or a meromorphic map between two compact, complex, manifolds)X and Y. Ordinarily one eliminates the indeterminacy locus ...
0
votes
0answers
115 views

birational invariants of projective surfaces

I am studying Castelnuovo's rationality criterion for surfaces. Let $S$ be a projective surface and $K$ a canonical divisor on $S$. Let's use the notation ...
4
votes
2answers
162 views

$Aut(X)$ and $Bir(X)$ for Calabi-Yau 3-folds with $\rho(X)=1$

Let $X$ be a Calabi-Yau 3-fold with Picard number one. How can one show that the automorphism group $Aut(X)$ is finite and moreover coincides with the birational automorphism group $Bir(X)$? It seems ...
2
votes
0answers
100 views

Global sections of exceptional divisor in normalized blow-up

Let $(R, \mathfrak{m})$ be a Noetherian normal local domain and $I$ an $R$-ideal. Write $X$ for the normalization of $\mathrm{Proj}(R[It])$ and $E$ for the effective Cartier divisor defined by the ...
1
vote
0answers
67 views

invariance of the dimension of severi varieties of surfaces

Suppose I have a smooth projective surface $S\subset P^n$ embedded by a very ample linear system $|L|$. Consider now the generalized Severi varieties that parametrize curves on $S$ belonging to $|L|$, ...
2
votes
1answer
134 views

Moduli spaces admitting birational morphisms over moduli spaces of curves

There are many alternative compactifications of $M_{g,n}$ which live naturally under the classical Deligne-Mumford compactification $\overline{M}_{g,n}$. For instace the moduli spaces of weighted ...
1
vote
1answer
144 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 ...
1
vote
0answers
104 views

stably birational abelian varieties are isomorphic

Can anybody help me to prove the following result: Proposition. Let $A$ and $B$ be abelian varieties over a field $k$ of characteristic zero. Assume that $A \times \mathbb{P}_k^n$ and $B \times ...
3
votes
1answer
130 views

Are weak Fano 4-folds with canonical Gorenstein singularities bounded?

A Fano variety over $\mathbb{C}$ with Gorenstein singularity is called weak Fano if the anti-canonical divisor is nef and big. Are there finite families of weak Fano 4-folds with canonical ...
3
votes
1answer
132 views

Is there a blow-up formula for the derived category of a singular ambient variety?

For a nonsingular variety sitting inside a nonsingular ambient variety there is a semi-orthogonal decomposition of the derived category of the blow-up (with center that subvariety). What can be said ...
2
votes
2answers
233 views

Rationally connected varieties and rational fibrations

Let $Y$ be a rationally connected variety over an algebraically closed field, and let $$\phi:X\dashrightarrow Y$$ be a rational fibration such that the general fiber of $\phi$ is rationally chain ...
2
votes
1answer
155 views

Tangent bundles and birational morphisms

Let $f:X\rightarrow Y$ be a smooth morphism between smooth schemes. Then there is an exact sequence $$0\mapsto T_{X/Y}\rightarrow T_{X}\rightarrow f^{*}T_{Y}\mapsto 0$$ Now let us assume $f$ to be a ...
4
votes
1answer
164 views

Étale covers and birationality of varieties

All varieties are assumed to be projective over $\mathbb{C}$. Let $f_1: Y \to X$ and $f_2: Y' \to X$ be ├ętale morphisms with same finite Galois groups (to be honest, I don't know what does Galois ...
2
votes
1answer
168 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 ...
1
vote
1answer
116 views

On Non F-pure ideal and Sharp F-Purity for a pair $(X, \Delta)$ where $K_X+\Delta$ is NOT $\mathbb{Q}$-Cartier

Suppose $(X,\Delta\ge 0)$ is a pair such that $(p^g-1)(K_X+\Delta)$ is an Integral Weil Divisor for some $g>0$ and $X$ is a normal variety. Define $\mathcal{L}_{e,\Delta} = \mathcal{O}_X( ...
5
votes
2answers
308 views

Hn(X, OX) = 0 for X birational to a regular affine variety?

It is a basic fact that $H^n(X, F) = 0$ if $X$ is noetherian affine, $n > 0$, and $F$ a quasi-coherent sheaf. If $Y \to X$ is a blow-up of a smooth variety in a smooth center, then then ...
9
votes
1answer
415 views

Why is proving fully-faithfulness of an integral functor locally analytically sufficient?

More than once I've come across a statement in a paper about derived categories in which it says something to the effect of "in order to prove that $\Phi:D^b(X)\rightarrow D^b(Y)$ is fully-faithful we ...
8
votes
1answer
194 views

Conditions for the contractibility of subvarieties

One often finds statements of the sort "and one can contract this subvariety $E\subset X$ to a point in the projective variety $W$," without any explanation of the reasons such a contraction exists, ...
4
votes
1answer
102 views

Are there only finitely varieties of general type dominated by a given variety in the following sense

Let $X$ be a smooth projective complex algebraic variety. I believe the answer to the following question should be well-known. It is an instance of the Iitaka conjecture. There are only finitely many ...
10
votes
2answers
443 views

Geometrically unirational varieties that are not unirational

By a variety over a field $k$, I mean a scheme that is separated and of finite type over $k$. I indicate changes of the base ring by subscripts. Does there exist a smooth and projective variety $V$ ...
6
votes
1answer
294 views

Proving a variety is not unirational

It is known that if a variety is unirational then it is rationally connected. However, there are no known examples of rationally connected varieties which are not unirational. In these notes, at the ...
2
votes
0answers
210 views

Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces

Is there any version of Kawamata-Viehweg vanishing theorem that holds for excellent surfaces? I will be very glad if there is one such, but if not then is it at least true for a surface over a non ...
3
votes
0answers
225 views

Adjunction Formula for Weil Divisors on a Normal Variety X

Let $X$ be a normal variety over an algebraically closed field $k$ of characteristic $p>0$ and $S$ be a prime Weil divisor on $X$ which is normal too. Now if $K_X+S$ is NOT $\mathbb{Q}$-Cartier, ...
1
vote
0answers
64 views

K-equivalence does not depend on the choice of the third variety

By K-equivalent of two smooth varieties $X,Y$, we mean there exist a smooth variety $Z$, and birational morphism $q: Z \to X,\quad p: Z \to Y$ , such that $q^* \omega_X \cong p^* \omega_Y$. Suppose ...
0
votes
0answers
171 views

How to Construct a ''Nice'' Birational Model in Characteristic p>0

Let $X=Spec\ A$ be a normal affine variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Also, assume that $S\subset X$ be a prime Weil-divisor on $X$. Now, I ...
3
votes
1answer
167 views

About del Pezzo surfaces

First of all, I would like to apologize if my question is stupid or a well known fact. Let $F$ be a rational surface with $K_F^2=5$ and $f: F\rightarrow \mathbb{P_k^2}$ be a birational morphism ...
1
vote
0answers
116 views

Is always a Crepant birational map between smooth varieties a small modification

Lemma: A crepant birational map between quasi-projective normal varieties with only terminal singularities is in fact small modifitacions, i.e. isomorphism in codimension 1. So, if ...
2
votes
1answer
155 views

Lifting vector fields to its resolution in char $p$

In this paper 4.7, the authors showed that on a normal variety $X$, if there is a tangent vector field on its smooth locus, then it can be lifted as a logarithmic tangent vector field on a log ...
2
votes
1answer
161 views

Crepant Birational Map on the Blow-up

Let $ f: \mathbb{P}^n \dashrightarrow (\mathbb{P}^1)^n , [x_0:\dots :x_n] \mapsto ([x_0,x_1],[x_0,x_2], \dots ,[x_0,x_n])$ a birational map. In particular, if $X$ is the blow-up of $\mathbb{P}^n$ at ...
1
vote
1answer
133 views

Crepant Birational Map

Let $ f: \mathbb{P}^n \dashrightarrow (\mathbb{P}^1)^n , [x_0:\dots :x_n] \mapsto ([x_0,x_1],[x_0,x_2], \dots ,[x_0,x_n])$ a birational map. Is $f$ a Crepant birational map?
2
votes
1answer
152 views

On a Strongly F-regular Pair (X, \Delta)

Let $X$ be a normal projective variety over a field of characteristic $p>0$ and $(X, \Delta\geq 0)$ be a pair such that $K_X+\Delta$ is $\mathbb{Q}$-Cartier whose index is not divisible by $p$. ...
4
votes
1answer
96 views

Strictness of the inequality relating the Iitaka dimension and algebraic dimension

For any (compact and connected) complex manifold $X$ and any line bundle $L$ on $X$ we have the well known inequalities $\kappa(X,L)\leq\alpha(X)\leq\dim(X)$ relating the Iitaka-dimension ...
9
votes
4answers
369 views

When does $Aut(X)=Bir(X)$ hold?

Let $X$ be a projective complex manifold. Under what condition do we have the equality $Aut(X)=Bir(X)$? Here $Aut(X)$ denotes the group of holomorphic automorphisms of $X$ and $Bir(X)$ the group of ...
6
votes
2answers
321 views

Quotients of rational surfaces

Let $X$ be a projective surface defined over a field $k$ of characteristic $0$, and let $G$ be a finite group acting biregularly on $X$. Assuming that $X$ is rational over $k$, is the quotient $X/G$ ...