Questions tagged [birational-geometry]

Birational geometry is a field of algebraic geometry the goal of which is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles.

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5
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1answer
186 views

Isomorphisms of complete intersections

Let $X, Y\in \mathbb{P}^n$ be two singular Fano complete intersections of the same multidegree $(d_1,…,d_r)$. If we assume there is an isomorphism $f\colon X\rightarrow Y$ are there any assumptions so ...
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1answer
79 views

Weak Fano varieties and small transformations

A projective normal and $\mathbb{Q}$-factorial variety $X$ is said to be log Fano if there exists and effective divisor $D$ on $X$ such $-K_X-D$ is ample and the pair $(X,D)$ is klt. Now, let $f:X\...
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1answer
128 views

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\...
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1answer
121 views

Cohomology of normal bundle and tangent bundle on Gushel-Mukai threefold

Let $X$ be a smooth general ordinary Gushel-Mukai threefold. There is an embedding $X\rightarrow\mathrm{Gr}(2,5):=G$. Consider the normal bundle $\mathcal{N}_{X|G}$, how to compute cohomology of this ...
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Divisorial contractions and singularities

I have a smooth $6$-fold $X\subset\mathbb{P}^n$ and a divisor $D\subset X$ cut out by a quadratic polynomial. I know that $D$ in singular along a smooth $3$-fold $Y\subset X$, and that if $Z$ is the ...
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133 views

Is there a rational projective hypersurface of degree three?

A variety $ Y $ of dimension $ n $ is rational if there is a birational map $ \phi: \mathbb{P}^{n}_{k} \dashrightarrow Y $. Does there exists a rational hypersurface of degree three?
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112 views

Dominant rational maps and compositions

According to many books (and also to Stack project, see https://stacks.math.columbia.edu/tag/01RI ), a morphism $f\colon X\to Y$ between schemes is said to be dominant if the image is dense in $Y$. ...
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The structure of the Hilbert scheme of conics contained in hypersurfaces in $\mathbb P^3$

We work over a field of characteristic $0$. Let $X\hookrightarrow\mathbb P^3$ be a geometrically integral hypersurface of degree $\delta$. It is well known that the Hilbert scheme of conics in $\...
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1answer
97 views

Kahler cone of blow up of $\mathbb{C}P^1 \times \mathbb{C}P^n$

What is the Kahler cone of $\mathbb{C}P^1 \times \mathbb{C}P^n$ blown-up along a co-dimension two subvariety of the form $pt \times H$ where $H \subset \mathbb{C}P^n$ is a hyperplane?
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1answer
135 views

Normal bundle and small contraction in threefolds

Let $f:X \to \mathbb{A}^1$ be a smooth, projective morphism of relative dimension $2$. Suppose that the fiber $X_0:=f^{-1}(0)$ contains an irreducible rational curve, say $C$ such that the restriction ...
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Is there a direct way to show Fano surface of lines and conics on the pairs of Fano threefolds isomorphic?

I am considering the following setting: Let $(Y_d, X_{4d+2})$ be the pair of degree $d$ and index 2 Fano threefold $Y_d$ and degree $4d+2$ index 1 Fano threefold and both of them are Picard number 1. ...
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1answer
106 views

Log resolution of a variety of log general type

Work over the complex numbers. Let $(B, \Delta)$ be a normal irreducible variety of log general type, i.e., $K_B + \Delta$ is ample. Let $f : (\widetilde{B}, \widetilde{\Delta}) \to (B, \Delta)$ be a ...
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139 views

Blow up along codimension 1 subscheme

Let $X$ be a smooth projective variety. $Z$ be a closed subscheme of codimension $1$ (potentially with embedded points). $Z_{red}$ is a Cartier divisor and $Bl_{Z_{red}}X$ is just $X$. What about the ...
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1answer
171 views

(stable)-base locus on fibres

Let $X\to Y$ be a morphism between projective varieties, with general fibre being smooth and $Y$ being a smooth curve. Let $D$ be a divisor on $X$. Is is true that for a general fibre $F$ and $m\ge 1$ ...
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When is the dualizing sheaf globally generated?

Let $X$ be a projective integral Cohen-Macaulay variety (over $\mathbb{C}$, if that makes things easier). The Cohen-Macaulay condition says that the dualizing complex (see this answer) is concentrated ...
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214 views

Non-uniruled connected smooth fibers implies flat

Let $f:X\to Y$ be a surjective morphism of connected smooth projective varieties over an algebraically closed field. Assume all fibers are connected smooth and none are uniruled. Is $f$ flat? In ...
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336 views

Variety of negative Kodaira dimension contains a projective line

Does a smooth projective variety over an algebraically closed field of negative Kodaira dimension contain a projective line? I do not want to assume any conjectures.
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132 views

Steps of the MMP “in family”

Let $\pi\colon X\to Y$ be a morphism between irreducible varieties with terminal singularities (let us say smooth if you want). I suppose that I have an open subset $U$ of $Y$ over which the fibres ...
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215 views

birational geometry of moduli spaces: why work on the coarse space?

In studying the birational geometry of $\overline{\mathcal{M}}_g$, it seems standard to work with the coarse space $\overline{M}_g$ rather than the smooth stack $\overline{\mathcal{M}}_g$. Why is this?...
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1answer
176 views

Multiplication maps for big line bundles

In Birational Geometry of Algebraic Varieties, Kollar and Mori write that for a line bundle "being big is essentially the birational version of being ample" (page 67). Recall that a line ...
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1answer
135 views

Decomposition of a morphism with positive dimensional fibers

It is well known that any birational morphism between projective varieties is a sequence of blow ups. Suppose now that I have a morphism $f:X \to Y$ with positive dimensional fibers, that is a ...
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Unirationality of $\mathrm{Sym}^2(C)$ of genus $3$ curve

Let $k$ be a field, let $C$ smooth genus $3$ curve, let $S:=\mathrm{Sym}^2(C)=C\times C/(\mathbb{Z}/2\mathbb{Z})$, it is a smooth surface of general type, therefore it is not unirational when $\mathrm{...
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1answer
179 views

How to find a rational $\mathbb{F}_{\!q}$-curve on a quite classical Calabi–Yau threefold?

Take a finite field $\mathbb{F}_{\!q}$ such that $q \equiv 1 \pmod 3$, i.e., $\omega \mathrel{:=} \sqrt[3]{1} \in \mathbb{F}_{\!q}$, $\omega \neq 1$. Also, for $i \in \{0,1,2\}$ consider the elliptic ...
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1answer
129 views

Extending rational maps of nodal curves

Let $R$ be a discrete valuation ring with fraction field $K$ and $C, D$ two nodal (=prestable) curves over $\operatorname{Spec} R$. If I have a map $C_K \to D_K$ between the restriction of the curves ...
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164 views

Which field extensions do not affect Chow groups?

Let $X$ be a (say, smooth projective) variety over a field $k$. For which $K$ it is known that the ("ordinary", that is, not higher) Chow groups of $X$ map onto that of $X_K$ bijectively? ...
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97 views

Whether a particular fiber product of varieties is integral

Let $X,Y,Z$ be irreducible projective varieties over $\mathbb{C}$ (you can assume all of them are normal), and let $f:X\rightarrow Z, g: Y\rightarrow Z$ be two birational projective morphisms, such ...
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2answers
190 views

Example of a projective variety over a field of characteristic zero which is uniruled but not ruled

A variety $ Z $ over a field $ k $ of characteristic zero is ruled if there is a variety $ M $ and a dominant, birational map $ \phi: M \times \mathbb{P}^{1}_{k} \dashrightarrow Z $. A variety $ Z $ ...
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1answer
213 views

An explicit negative solution to the Lüroth problem for non-algebraically closed fields

Let $\mathsf{k}$ be a field of characteristic $0$, and consider $\mathsf{k}(x,y)$. If $\mathsf{k}$ is algebraically closed, then every field $L$ such that the inclusion $\mathsf{k} \subset L \subset \...
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114 views

An analogue of Noether's Problem for non-rational varieties

For the sake of simplicity, let $\mathsf{k}$ be algebraically closed and of zero characteristic. Varieties are irreducible. The (linear) Noether's Problem (which goes back to the early 1910's in ...
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112 views

Log Calabi-Yau variety diffeomorphic to an algebraic torus

Let $U$ be a complex affine log Calabi-Yau variety, which I take to mean a smooth affine variety which admits a compactification in a smooth projective variety $X$ with an snc anti-canonical ...
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189 views

Wonderful compactification of $\mathrm{SL}(2)/\mathrm{SO}(2)$

Let $\mathbb{P}^2 = \mathbb{P}(\operatorname{Sym}^2\mathbb{C}^2)$ be the projective space of $2\times 2$ symmetric matrices over $\mathbb{C}$ modulo scalar. Define an $\mathrm{SL}(2)$-action on $\...
14
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1answer
281 views

Degree of secant varieties of Veronese varieties

Consider the degree two Veronese embedding $\mathbb{P}^n\rightarrow\mathbb{P}^N$ and let $V^n_{2}\subset\mathbb{P}^N$ be the corresponding Veronese variety. Let $Sec_k(V^n_{2})\subseteq\mathbb{P}^N$ ...
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128 views

Clemens-Griffiths component birational invariant

Let $X$ be a smooth variety over complex numbers $\mathbb{C}$, say a threefold for sake of better intuition. Is there any geometrical intuition behind the fact that the Clemens-Griffiths component of ...
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113 views

Realising flips as quotients of flops by an involution

In Séminaire BOURBAKI, 4ième année, 1988-89, n° 712, Juin 1989, Kóllar wrote the beautiful article Minimal Models of Algebraic Threefolds: Mori's program. Firstly, I would highly encourage this ...
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96 views

Determining if a morphism is a blowup along a given subvariety

Let $X,\tilde{X}$ be two smooth projective varieties over $\mathbb{C}$, and let $\pi:\tilde{X}\rightarrow X$ be a projective morphism. Let us moreover assume that there exists a smooth closed ...
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1answer
152 views

Certain endomorphisms of $\mathbb{C}(x,y)$

Let $f: (x,y) \mapsto (p,q)$ be a $\mathbb{C}$-algebra endomorphism of $\mathbb{C}(x,y)$ satisfying the following two conditions: (i) $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$. (...
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116 views

Lefschetz type theorems for linear sections

Let $X\subset\mathbb{P}^n$ be e normal variety, $L\subset\mathbb{P}^n$ a linear subspace, and $Y = X\cap L$ a linear section. Assume that $Y$ is also normal. In particular, we have that $Sing(X)$ has ...
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1answer
209 views

A property of varieties between unirational and retract rational

EDIT: The vague question Q1 below is partially answered, while the concrete question Q2 seems to be still open. Let $V$ be a geometrically integral variety over a field $K$. I consider the following ...
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69 views

Picard numbers of isogenous K3 surfaces over a non-closed field

Let $S_1, S_2$ be K3 surfaces defined over a field $k$ and $\phi\!: S_1 \dashrightarrow S_2$ a dominant rational $k$-map (so-called isogeny). It is known that $\rho(S_1) = \rho(S_2)$ for the complex ...
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104 views

Does nefness carry over through flips?

Suppose $\pi: X \dashrightarrow Y$ is a birational map which is an isomorphism in codimension 1 (such as a flip). Also suppose both $X$ and $Y$ have reasonable (say log terminal) singularities. We ...
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136 views

What is the exceptional divisor of a divisorial contraction?

Let $\pi \colon X\to Y$ be a divisorial contraction between $\mathbb{Q}$-factorial terminal varieties. Let $E\subset X$ be the exceptional divisor and let $\pi(E)=D$ be its image. It is true that $\...
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1answer
94 views

Mori extremal contraction with small Betti number

Is there example of a smooth, projective, complex $3$-fold $X$, having $b_{2}(X)=2$ a Mori extremal contraction $\phi: X \rightarrow X'$ which contracts a smooth quadric surface $Q \subset X$? It ...
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1answer
101 views

Cremona transformation $\sigma: \mathbb{P}^2 \dashrightarrow \mathbb{P}^2 $ and pushforward of divisor

Let consider the birational Cremona transformation $\sigma: \mathbb{P}^2 \dashrightarrow \mathbb{P}^2 $ defined by $$(X:Y:Z) \mapsto (X^{-1}:X^{-1}: Z^{-1})=(YZ:XZ:XY)$$. In language of birational ...
2
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1answer
177 views

Picard group modulo codimension 2

Let $X$ be a normal (possibly singular) projective surface over $\mathbb{C}$. Consider the set $M_X$ of all coherent sheaves $F$ on $X$ such that there exists a finite subset $Y\subset X$ such that $F$...
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1answer
120 views

Is $\mathbb{Q}$-factoriality preserved under contraction?

Let ($X$,$\Delta$) be projective klt pair and $f \colon X \rightarrow Z$ be contraction of ($K_X + \Delta$) - negative extremal ray $R$. If $X$ is $\mathbb{Q}$ -factorial and $\mathrm{dim}Z < \...
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108 views

Isomorphisms of weighted complete intersections

Let $X\subset\mathbb{P}(a_0,\dots,a_n)$ and $Y\subset\mathbb{P}(b_0,\dots,b_n)$ be two weighted complete intersections with mild (say terminal) singularities. Assume that there is an isomorphism $f:...
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142 views

Jacobian fibration of an abelian fibration

Let $f \colon S \rightarrow C$ be a minimal elliptic surface and let $g \colon J \rightarrow C$ be its jacobian fibration. In this case, we know that the fibers of $g$ are better behaved that the ones ...
4
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1answer
106 views

Terminal $\mathbb{Q}$-factorial divisorial contractions

Let $X$ be a $3$-fold, and $f:Y\rightarrow X$ a birational $\mathbb{Q}$-factorial divisorial terminal contraction (of relative Picard number one) contracting a divisor $E\subset Y$ to a point $p\in X$....
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73 views

Concerning $\mathbb{C}(s_1,s_2,s_3,y)=\mathbb{C}(x,y)$, where $s_1,s_2,s_3$ are symmetric

Perhaps the following question is not in the level of MO questions, but it has not received comments in MSE, so I ask it here also: Let $\beta: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$ be the involution ...
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1answer
144 views

On the birational equivalent class of algebraic surfaces with Picard number $1$

An open subset $U$ of a projective surface $Z$ is big if $\mathrm{codim}_Z(Z\setminus U)\geq2$. Let $X$ and $Y$ be smooth complex projective surface. If there exists a birational map $f:X\...

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