# Tagged Questions

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### $\mathbb{P}^1$-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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $Y$?
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### 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 ...
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### $Aut(X)$ 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 ...
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### 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 ...
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### 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 $|L|$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 $V$ ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 $p$. ...
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### 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 ...
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### 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 ...
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### 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 $X/G$ ...
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### Training towards research on birational geometry/minimal model program

Being a not yet enrolled independently supervised graduate student in mathematics, with prospects of applying to American graduate schools hopefully in a 1-2 years' time, I have a background of having ...
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### Fields over which cubic hypersurfaces are rational

All cubic hypersurfaces having at least one double point are birational to some $P^n$ over an algebraically closed field. How does the statement change as I pass to non alg closed fields? Does it hold ...
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### is Hasse principle a birational invariant?

...it is probably a very trivial question, but I am a beginner in arithmetics.
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### Numerical dimension of nef divisors

Let $D$ be a nef divisor (moreover suppose it is effective if you prefer) on a normal projective variety of dimension $n$. Let $k\in[1,n-1]$. If $D^k\cdot V=0$ for generic subvarieties $V\subseteq X$ ...
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### Birational maps and open subsets inducing isomorphisms

Let $\pi \colon X\to Y$ be a morphism, where $X$ is projective. Suppose the existence of an open subset $U\subset X$ such that $\pi^{-1}(\pi(U))=U$, that $\pi$ is injective on $U$, and that its ...
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### Crepant resolutions of ODP's on a 3-fold

It seems to be a well-known fact that if we have a 3-fold $X$ with only ODP singularities (ordinary double point) and a smooth Weil divisor $E$ passing through them, then by blowing-up $X$ along $E$ ...
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### Extending birational isomorphisms between planar curves to the P^2

Let $k$ be a field and let $C,D$ be two integral curves in $\mathbb{P}^2_k$. Now let $f:C \to D$ be a birational isomorphism. Can $f$ be extended to $\mathbb{P}^2_k$. To be precise, does there exist a ...
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### Discussion of Luroth's problem in an article of Beauville

I am reading a wonderful article of Arnaud Beauville, called La théorie de Hodge et quelques applications http://math.unice.fr/~beauvill/conf/Bordeaux2.pdf There is one place on page 12 that I can ...
Let $\pi:X\to Y$ be a proper map of algebraic varieties (over $\mathbb C$) which is a bi-rational equivalence. are the following statements equivalent? The derived direct image of $O_X$ is $O_Y$. ...
### Why is $\mathbb{Q}$-factoriality not local in the étale topology?
I was reading Kollár and Mori's book today and stumbled on the following passage: "The $\mathbb{Q}$-factoriality assumption is a very natural one if we start with smooth varieties, and it makes many ...