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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Is there a classification of minimal algebraic threefolds?

The minimal model program aims to find a minimal representative in the birational class of a given variety with reasonable singularities. Assuming this has been done, it seems natural to ask what ...
Kim's user avatar
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Birationally equivalent elliptic curves and singularities

I got the following cubic elliptic curve from some physical problem $$E_c(\mathbb{C}): w^2=4 z^3-zG_2-G_3,$$ where $G_2=3 \alpha ^2+\gamma$ and $G_3=\alpha ^3-\alpha \gamma -\beta ^2$ for known ...
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2 votes
1 answer
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Question regarding linear system of projective space

I am currently reading the paper titled "Birational Geometry of Moduli spaces of Configurations of Points on the Line" by M.Bolognesi and A.Massarenti. I have following doubts in section 2....
tota's user avatar
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Is there any relation between the pushforward of a divisor and the pushforward of its line bundle?

Given a morphism between normal varieties $f: X \to Y$, we can push forward a Cartier divisor $D$ to get a cycle $f_* D$. On the other hand, we can form the line bundle $\mathscr{L}(D)$ and push that ...
Kim's user avatar
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6 votes
1 answer
546 views

Different definition of Cox rings

Definition: Let $X$ be a normal projective variety with finitely generated Picard group. Define the Cox ring of $X$ as the multisection ring $$\text{Cox}(X)=\bigoplus_{(m_1,\ldots,m_k)\in \mathbb{N}^k}...
Plano's user avatar
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2 votes
1 answer
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Sheaves on families of genus 2 curves in Hassett's paper

Sorry for a maybe stupid long question but I'm reading the paper "Classical and minimal models of the moduli space of curves of genus two" by Brendan Hassett and I'm not able to unravel a ...
gigi's user avatar
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5 votes
1 answer
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Equivalent definitions of Kodaira dimension

The Kodaira($-$Iitaka) dimension of a line bundle $L$ on a complex manifold $X$ can be defined either in three ways: The maximal dimension of the image of the rational maps $φ_{|mL|} : X \...
Carlos Esparza's user avatar
1 vote
1 answer
158 views

Positivity of the global log canonical threshold of a pair

Let $(X,L)$ be a polarized smooth projective variety. Let $D$ be a smooth irreducible divisor in $X$. Let $0<c<1$ be a real number. We denote $cD$ as $\Delta$. We can define the $\alpha$ ...
Chenxi Yin's user avatar
6 votes
1 answer
692 views

Singular curves of genus 1

Let $C$ be an irreducible curve of arithmetic genus $1$ over a field $k$ and with a double $k$-point $p\in C$. Is $C$ rational over $k$? If $C$ is a plane cubic the answer is positive since we can ...
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4 votes
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Volume of conic bundles

Consider a smooth conic bundle $X\rightarrow \mathbb{P}^1$ with discriminant of degree $d$ (the locus of $\mathbb{P}^1$ over which the fibers are reducible conics). There is a formula for $(-K_X)^2$ ...
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Confusion about the (Grothendieck–Poincaré) double dual of reflexive differentials vs usual differentials on a normal Cohen–Macaulay scheme

$\DeclareMathOperator\Hom{Hom}$Let $\mathcal{A}$ be an abelian category, my question is about the case when $\mathcal{A}$ is the category of quasi-coherent sheaves on a scheme $X$. There is a fully ...
Adam's user avatar
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4 votes
2 answers
480 views

Smoothness of fibers over finite fields

Let $f:X\rightarrow Y$ be a morphism of smooth projective varieties over a finite field of characteristic different from $2$. Is there any result on the existence of a point $y\in Y$ such that $X_y = ...
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2 votes
0 answers
152 views

Does anyone know a rationally chain connected, Cohen Macaulay variety which is not separably rationally connected?

An $ n $-dimensional variety (here variety means an integral, separated, scheme of finite type over an algebraically closed field) $ X $ over a field $ k $ is rational if there is a birational map $ \...
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Del Pezzo surfaces of degree four and complete intersections of two quadrics

Let $X = Q_1\cap Q_2$ be a complete intersection of two smooth quadrics, over a field $K$, in $\mathbb{P}^4$ with homogeneous coordinates $y_0,y_1,y_2,y_3,y_4$. Set $Q_1 = \{F_1 = 0\}$ and $Q_2 = \{...
Puzzled's user avatar
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6 votes
2 answers
675 views

Smooth complete intersections

Let $X_{2,3}\subset\mathbb{P}^n$, with $n\geq 5$, be a complete intersection of a quadric $X_2$ and a cubic $X_3$ containing a $2$-plane $H$. Assume $X_2$ and $X_3$ to be general among the ...
Puzzled's user avatar
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Why do such a birational map exists? And why it is unique?

Let $G$ be a complex linear algebraic group which is connected and reductive and let $\mathfrak g$ be its Lie algebra. Suppose that $H \subset G$ is a 1-dimensional torus such that the action of $H$ ...
Bobech's user avatar
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1 answer
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Space of rational conics

Let $K$ be a field of characteristic different from two. Conics over $K$ (that is curves of degree two in $\mathbb{P}^2_K$) are parametrized by $\mathbb{P}(k[x,y,z]_2) = \mathbb{P}^5_K$. Conisider the ...
Puzzled's user avatar
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2 votes
1 answer
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Linear subspace in quadric hypersurfaces over a field

Let $K$ be a field of characteristic different from two, and $Q\subset\mathbb{P}^{n+1}_K$ an $n$-dimensional smooth quadric hypersurface over $K$. Suppose also that $Q$ has a $K$-point and so $Q$ is ...
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-3 votes
1 answer
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Show that a polynomial of degree 4 is birational equivalent to a polynomial of degree 3 [duplicate]

Suppose that $f_{4}(x)$ is a polynomial of degree 4 with no multiple roots, $C$ is the curve defined by $y^{2}=f_{4}(x)$, I want to show that there is a polynomial $f_{3}(x)$ of degree 3 with no ...
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126 views

Dominant rational map and linear systems

Let $X$ be a (smooth) projective variety of dimension $n$ and $L$ be an invertible sheaf on $X$ such that there is a linear subspace $V \subset H^0(L)$ of dimension $n+1$ such that the associated ...
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5 votes
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Rational contraction and Proj of section ring

I am reading the paper "Mori Dream Spaces and GIT" by Hu and Keel. https://arxiv.org/abs/math/0004017 I cannot understand the proof of Lemma 1.6 in it. Let $X$ be a normal projective variety....
Kensuke_Yamato's user avatar
4 votes
1 answer
257 views

When is the birational Torelli problem for CY threefolds true?

I am aware from Borisov, Căldăraru, Perry and Ottem, Rennemo that what is known as the birational Torelli problem is false in general for Calabi-Yau threefolds, but I would like to know if there are ...
AT0's user avatar
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1 vote
2 answers
410 views

Embedding of a blow-up

In $\mathbb{P}^1\times\mathbb{P}^2$ take a general divisor $X$ of type $(0,2)$. Consider two general divisors $H_1,H_2$ of type $(2,1)$ and set $Y = X\cap H_1\cap H_2$. Let $Z$ be the blow-up of $X$ ...
Puzzled's user avatar
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1 vote
0 answers
227 views

Hironaka's construction for compact Kähler manifolds

In Hartshorne's book 《Algebraic Geometry》 p.443, the author introduces a construction of a non-projective complex manifold from a projective one. His method can be summarized as following: Let $X$ be ...
Tom's user avatar
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2 answers
548 views

Birational geometry over finite fields

I apologize in advance since probably my questions are very naive. I would like to understand some central notions in birational geometry, that are clear to me over the complex numbers, for varieties ...
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3 votes
1 answer
168 views

Embedding quadric bundles

Let $\pi:X\rightarrow W$ be a morphism of smooth projective varieties over a field $k$ whose generic fiber is a smooth quadric, and let $r$ be the dimension of the fibers of $\pi$. Does there always ...
Puzzled's user avatar
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2 votes
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Rational points on surfaces

Let $k$ be a field of characteristic zero. In the affine space $\mathbb{A}_{x,y,t}^3$ consider a surface $S$ of the form $$ S = \{a_0(t)x^2+a_1(t)xy+a_2(t)x+a_3(t)y^2+a_4(t)y+a_5(t) = 0\} $$ where $...
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1 vote
0 answers
281 views

Questions about Hironaka's example

In Hartshorne's book 《Algebraic geomery》 p.443, the author gives an explanation of Hironaka's example on non-Kähler deformation of compact Kähler manifolds, his construction can be summarised as ...
Tom's user avatar
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4 votes
2 answers
741 views

Are Du Val singularities smoothable?

I am interested in when a Du Val surface singularity is smoothable. By Du Val singularity, I mean (the germ of a) isolated double point surface singularity admitting a resolution by blowups of ...
Evgeny T's user avatar
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7 votes
0 answers
300 views

Number of rational points over finite fields mod $q$ is birational invariant

I heard that if $\mathbf F_q$ is a finite field, $X, Y$ are birational smooth proper variety over $\mathbf F_q$, then $\#(X(\mathbf F_q)) \equiv \#(Y(\mathbf F_q)) \pmod q$, and I heard that the proof ...
Aoi Koshigaya's user avatar
2 votes
1 answer
191 views

Existence of terminal $3$-fold flips

Does there exist a terminal $3$-fold $X$ with a curve $C\subset X$ such that $K_X\cdot C < 0$ admitting a Mori flip $X\dashrightarrow Y$, flipping $C$ to a curve $C'\subset Y$, where the singular ...
Puzzled's user avatar
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5 votes
1 answer
348 views

diagonal cubic hypersurfaces

At the end of https://encyclopediaofmath.org/index.php?title=Cubic_hypersurface#References it is stated that the diagonal cubic hypersurface $$ \sum_{i=0}^{2m+1} a_i x_i^3 = 0, m\ge 2 $$ (and ...
W Sao's user avatar
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4 votes
0 answers
127 views

Parameter spaces for conic bundles

A conic bundle over $\mathbb{P}^n$ is a morphism $\pi:X\rightarrow\mathbb{P}^n$ with fibers isomorphic to plane conics. A conic bundle $\pi:X\rightarrow\mathbb{P}^n$ is minimal if it has relative ...
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1 vote
0 answers
149 views

What's the modification of a Calabi-Yau manifold?

Recall that a modification of a compact manifold $X$ is a holomorphic map $\mu:\tilde X \to X$ such that: i) dim $\tilde X$=dim $X$; ii) there exists an analytic subset $S\subset X$ of codimension $\...
Tom's user avatar
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9 votes
0 answers
370 views

How can I "see" that a map is birational?

This came up with the Euler brick. Let $T=(p,q,r)$ be a Randall triple, i.e. $$(p^2-1)(q^2-1)(r^2-1)=8pqr\ \qquad\text{[eq.1]}.$$ There are tons of maps that map a triple $T$ to another $T'=(p',q',r')$...
Hauke Reddmann's user avatar
12 votes
1 answer
564 views

Does a resolution of a rational singularity have rationally connected fibers?

A rational singularity is a singularity of a complex variety $X$ such that for any resolution $\pi:\; \tilde X\rightarrow X$ the higher direct images $R^i\pi_*(O_{\tilde X})$ vanish for all $i>0$. ...
Misha Verbitsky's user avatar
4 votes
0 answers
245 views

Minimal $b_2$ in Sarkisov's construction

In the paper On the structure of conic bundles. Math. USSR, Izv., 120:355–390, 1982, Theorem 5.10, Sarkisov constructed the first example of non-rational, rationally connected $3$-fold $X$ with $H^{3}...
Nick L's user avatar
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3 votes
1 answer
382 views

Is the automorphism group of a normal affine scheme a group scheme or an algebraic space?

If $ \operatorname{Spec}(A) $ is a smooth affine scheme over an algebraically closed field $ k $, then is $ \operatorname{Aut}(\operatorname{Spec}(A)) $ a group scheme or an algebraic space? Please ...
schemer's user avatar
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1 vote
0 answers
196 views

Mori cone of Picard rank two varieties

Let $X$ be a smooth projective variety of Picard rank two. Assume that there exists a surface $S\subset X$ such that $$i^{*}:\text{Pic}(X)\rightarrow\text{Pic}(S)$$ is an isomorphism, where $i:S\...
Puzzled's user avatar
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1 vote
1 answer
166 views

Moduli spaces of horizontal curves

Let $f:X\rightarrow Y$ be a morphism of projective varieties. We may assume that $X$ and $Y$ are smooth, and $f$ is flat of relative dimension one. Fix an ample divisor $A$ on $X$. I would like to ask ...
Puzzled's user avatar
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3 votes
0 answers
199 views

Dynamical degree and spectral radius

Let $X$ be a smooth, projective surface over an algebraically closed field $k$ of characteristic zero, and let $f \in \mathrm{Bir}(X)$ a birational map. Let's denote $f_{\ast} : \mathrm{NS}(X) \...
Federico Barbacovi's user avatar
2 votes
1 answer
124 views

Cremona transformations and divisors

Let $L$ be an ample line bundle in $\mathbb{P}^n$, with at least $n$ global sections. Choose two sets of $n$ linearly independent global sections of $L$, say $S_1:=\{D_1,...,D_n\}$ and $S_2:=\{E_1,.......
Ron's user avatar
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2 votes
1 answer
408 views

Divisors on projective bundles

Let $\pi:X = \mathbb{P}(\mathcal{E})\rightarrow\mathbb{P}^n$ be a projective bundle, where $\mathcal{E}$ is a rank two vector bundle over $\mathbb{P}^n$. If $n = 0$ then $X = \mathbb{P}^1$, and for $n ...
Puzzled's user avatar
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-1 votes
1 answer
242 views

Coefficients of elliptic curves over function fields

Consider the projective plane $\mathbb{P}^2_{\overline{\mathbb{C}(t)}}$ over the algebraic closure of the function field $\mathbb{C}(t)$. Take the point $p_0 = [0:1:0]\in \mathbb{P}^2_{\overline{\...
Puzzled's user avatar
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3 votes
0 answers
304 views

Does every Fano variety contain every abstract curve?

It is a famous result of Mori that all Fano varieties (in characteristic $0$) contain rational curves. What if we replace rational curve with a specific curve of positive genus? Question. Is it true ...
azaha89's user avatar
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2 votes
0 answers
401 views

On the exceptional divisor of the resolution of indeterminacy locus of rational map

Let $f:X \dashrightarrow Y$ be a rational map between smooth, projective varieties over $\mathbb{C}$. We know that there is a resolution of the indeterminacy locus using which we obtain a smooth, ...
Jana's user avatar
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3 votes
1 answer
252 views

On tensor product of field extensions

Let $K$ be a field which is a (transcendental) extension of $\mathbb{C}$. Let $L_1, L_2$ and $M_1, M_2$ be two field extensions of $K$ (not necessarily algebraic) such that $$L_1 \otimes_K L_2 \cong ...
Chen's user avatar
  • 1,583
2 votes
1 answer
232 views

Blow-ups of surfaces over a field

Let $S$ be a smooth projective surface of Picard rank $\rho(S)$ over a field $K$, and $\overline{S}$ its algebraic closure. Take a point $p\in\overline{S}$ and denote by $\overline{X}$ be blow-up of $\...
user avatar
5 votes
2 answers
498 views

Divisors whose restriction is big

Let $f:X\rightarrow Y$ be a flat morphism of smooth projective varieties, and $\mathcal{L}$ an effective and ample line bundle on $Y$. For a divisor $A\in H^0(Y,\mathcal{L})$ set $X_A := f^{-1}(A)$. ...
user avatar
2 votes
1 answer
276 views

Construction of Jacobian Ideal

In Qing Liu's Algebraic Geometry and Arithmetic Curve, we have the following proposition(6.3.13): Let $S$ be a locally Noetherian scheme. Let $X$, $Y$ be smooth schemes over $S$. Then any immersion $f:...
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