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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33 views

Rational points on fibrations

Let $S\rightarrow\mathbb{P}^1$ a smooth surface over a field $k$ with a fibration over $\mathbb{P}^1$ whose fibers are either isomorphic to $\mathbb{P}^1$ or to the union of two $\mathbb{P}^1$'s ...
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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 ...
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Rationality of quadric bundles

Let $\pi:X\rightarrow W$ be a flat morphism of smooth projective varieties over a field $k$ whose generic fiber is a smooth quadric. Assume that $W$ is rational and denote by $n$ the dimension of $W$ ...
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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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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 ...
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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 ...
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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 ...
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133 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 ...
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273 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 ...
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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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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 $\...
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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 $T'=(p',q',r')$...
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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 $i>0$. ...
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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}...
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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 ...
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151 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\...
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150 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 ...
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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) \...
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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,.......
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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 ...
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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{\...
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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 ...
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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, ...
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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 ...
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161 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 $\...
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381 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 $X_A := f^{-1}(A)$. ...
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160 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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Fibers of period map

Consider the period map for some smooth varieties, $p:\mathcal{X}\rightarrow\mathcal{A}:X\mapsto J(X)$, where $\mathcal{X}$ is moduli space of certain varieties and $\mathcal{A}$ is moduli space of ...
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How to distinguish the singularities on moduli space?

Let me start with concrete examples. Let $X$ be a smooth special Gushel-Mukai threefold and $\mathcal{C}(X)$ be its honest Fano surface of conics, it has two irreducible components $\mathcal{C}(X)=\...
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Curves in conic bundles

Consider a smooth minimal $3$-fold conic bundle $f:X\rightarrow\mathbb{P}^2$. Then $X$ has Picard rank two and consequently also the vector space of $1$-cycles is $2$-dimensional. Then the cone of ...
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Motivation for birational geometry

I'm interested in how do people that work in birational geometry view their field — specifically, what are the kinds of geometric questions (as opposed to commutative-algebraic questions) that ...
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Simultaneous Log resolutions for both varieties and divisors

Let $X$ be a normal variety and $D \subset X$ be a prime divisor which is also normal. It is well-known that we can take a resolution $f: W \to X$ of $X$ such that $$\DeclareMathOperator{\Supp}{\...
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Global sections of multiples of a divisor

Let $D$ be an integral divisor on a smooth projective variety $X$. Consider the multiples $mD$ of $D$ for $m\geq 0$. Clearly, $h^0(X,mD) = 1$ for $m = 0$. Is there any example where $h^0(X,mD) = 0$ ...
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208 views

A question on effective divisors

Let $X$ be a projective variety with two morphisms $f:X\rightarrow Y$ and $g:X\rightarrow Z$ with irreducible fibers of positive dimension. Assume that $Pic(X) = f^{*}Pic(Y)\oplus g^{*}Pic(Z)$. Then ...
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165 views

Nef and pseudo-effective divisors over non algebraically closed fields

Let $X$ be a projective variety over a field $K$. As a consequence of Kleiman's criterion, when $K$ is algebraically closed, we have that if $D$ is a nef divisor on $X$ then $D$ is pseudo-effective. ...
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On the b-nefness of the moduli part of canonical bundle formula

I have recently been wondering about the existence of a canonical bundle formula in the following situation and am not sure how to proceed. Suppose $(X,B) \xrightarrow{f} Y$ is a fibration where $ (X,...
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Canonical covering stack of a flop

In section 6 of Kawamata's paper https://arxiv.org/abs/math/0205287, he defines the canonical covering stack. In the proof of theorem 6.5, he considered a flop $$X\xrightarrow{\phi}W\xleftarrow{\psi}Y$...
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183 views

Pseudoeffective divisors on surfaces

Consider a minimal smooth conic bundle $S$ of dimension two. Assume that there are two curves $C,F$ on $S$ such that $C^2 < 0$ and $F^2 = 0$. Let $D$ be a pseudoeffective divisor on $S$ such that $...
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Examples of complex manifolds with trivial Néron–Severi group?

$\DeclareMathOperator\NS{NS}\DeclareMathOperator\Pic{Pic}$Let $X$ be a compact complex manifold, assume projective if you'd like. Define the Néron–Severi group to be the quotient $$\NS(X) = \Pic(X) / \...
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Log canonical surface with an elliptic singularity

I would like to know if there is an example as follows: $X$ is a log canonical surface and $x \in X$ is an elliptic singularity such that The minimal resolution of $x$ is a circle of rational curves (...
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1answer
127 views

Restriction of small transformations

Let $\phi:X\dashrightarrow Y$ be an elementary small transformation (isomorphism in codimension $1$) between normal and $\mathbb{Q}$-factorial projective varieties. Then there are small contractions $...
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184 views

descent of nef divisors

Let $f:X \rightarrow Y$ be a flat surjective morphism of complex projective varieties with connected fibers, $X$ normal, $Y$ smooth and $dim (Y) < dim(X)$. Suppose $L \in Pic (X)$ such that $L|_{X_{...
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1answer
177 views

Rationality in pencil of projective varieties

Let $\pi: \mathcal{X} \to \mathbb{P}^1$ be a pencil of projective $\mathbb{C}$-varieties such that a general fiber is smooth. Let $\mathbf{P}$ be one of the properties: rational, unirational, stably ...
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153 views

Rational and rationally chain connected surfaces

A projective variety $X$ over the complex numbers is rationally connected if two general points of $X$ can be joined by a rational curve in $X$, and rationally chain connected if two general points of ...
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1answer
217 views

Demailly Campana Peternell Conjecture for isolated singularities

I have asked this question on StackExchange but didn't get an answer, therefore I am asking again here. If $M$ is smooth, and $T^*M\to$ Spec$H^0(T^*M,\mathcal{O}(T^*M))$ is a projective birational map,...
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2answers
178 views

Subsets of $\mathbb{N}$ arising as genera of smooth projective curves in a variety

Given a smooth projective variety, the genera of the smooth projective curves in it form a subset of $\mathbb{N}$. Assuming the dimension is at least $2$, I think this subset is polynomially spaced, i....
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81 views

Cone and contraction theorems for certain sub-klt pairs

Suppose $(Y,B_Y)$ is a sub-klt pair which is a birational model of a klt pair $(X,B)$: there exists a birational morphism $\pi:(Y,B_Y) \rightarrow (X,B)$ such that $\pi^{*}(K_X+B)=K_Y+B_Y$. We know ...
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1answer
126 views

Identifying plane scrolls

In this paper it is shown (Corollary 1.9) that if for a 3-dimensional variety $X\subset \mathbb{P}^r$ there is an open set of hyperplanes $Y\in(\mathbb{P}^r)^{\vee}$ such that $\forall H, H'\in Y$ the ...
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232 views

An invariance property of rational singularities

Let $X$ be a normal variety over a field of characteristic zero with rational singularities. If $\pi:Y \to X$ is a birational proper morphism with $Y$ also normal, then does $Y$ also have rational ...
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1answer
238 views

Pullback of $\mathbb{R}$-Cartier divisors

I am reading the recent book by Kawamata, Algebraic Varieties: Minimal Models and Finite Generation. There is an English translation here . In the bottom of page 16 he says that an $\mathbb{R}$-...

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