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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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\...
1 vote
1 answer
623 views

Are terminal singularities $ \mathbb{Q}$-factorial?

The proof of Lemma 5-1-5 in this 1987 paper by Kawamata, Matsuda and Matsuki (link on Projecteuclid) seems to say that a variety with terminal singularities is $\mathbb{Q}$-factorial ( I only need ...
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 ...
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) \...
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,.......
3 votes
2 answers
865 views

Rationality of conic bundles

Let $\pi:X\rightarrow\mathbb{P}^2$ be a $3$-fold conic bundle, and let $\Delta\subset\mathbb{P}^2$ be its discriminant. Assume that both $X$ and $\Delta$ are smooth and that $deg(\Delta)\geq 6$. Can ...
19 votes
2 answers
2k 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 $Y$?
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 ...
-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{\...
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 ...
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, ...
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 ...
2 votes
1 answer
1k views

Relative form of Kodaira's lemma?

If $X$ is a smooth projective variety, Kodaira's lemma states that a big line bundle $D$ can be decomposed (as $\mathbb Q$-divisors) as $A+E$, with $A$ ample and $E$ effective. I am wondering what ...
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 $\...
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 $X_A := f^{-1}(A)$. ...
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:...
2 votes
0 answers
218 views

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 ...
3 votes
0 answers
135 views

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)=\...
0 votes
1 answer
173 views

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 ...
1 vote
0 answers
121 views

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}{\...
6 votes
2 answers
480 views

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 $h^0(X,mD) = 0$ ...
2 votes
1 answer
418 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 ...
2 votes
1 answer
372 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. ...
0 votes
1 answer
253 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 $...
2 votes
0 answers
206 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_{...
1 vote
0 answers
66 views

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,...
1 vote
0 answers
90 views

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$...
3 votes
1 answer
460 views

How can every divisor be reached by a sequence of blow-ups?

The following is a result of Zariski [cf. Lemma 2.45 of Birational Geometry of Algebraic Varieties]. $X$ : an algebraic variety over a field $k$. $(R,m)$ : a DVR of the quotient field $K(X)$ ...
3 votes
1 answer
465 views

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) / \...
3 votes
0 answers
157 views

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 (...
3 votes
1 answer
187 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 ...
3 votes
1 answer
259 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,...
1 vote
0 answers
297 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 ...
3 votes
2 answers
182 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....
2 votes
0 answers
105 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 ...
3 votes
1 answer
151 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 ...
7 votes
1 answer
1k views

Crepant resolutions of cDV singularities?

Compound Du Val 3-fold singularities form a good class of singularities in 3-fold singularity theory. I would like to know which singularities admit crepant resolutions. If I remember correctly, $cA_{...
8 votes
1 answer
297 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 ...
3 votes
1 answer
389 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 $\mathbb{R}$-...
3 votes
1 answer
306 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 ...
1 vote
0 answers
167 views

Domain of definition of a rational map is determined by several coherent sheaves of ideals

I am reading the Hironaka's paper on desingularisation. One part of his work is about eliminating the indeterminacy of proper rational map. He says that this comes from the Principalization of Ideal ...
1 vote
1 answer
349 views

A question on linear projection of a smooth projective variety

Let $X$ be a smooth, projective $\mathbb{C}$-variety of dimension $n$. Fix a closed point $x \in X$ and an embedding of $X$ in $\mathbb{P}^m$ for some integer $m$. For a given $d$, denote by $\sigma_d ...
1 vote
1 answer
657 views

Algebraic closure of $\mathbb{C}(t)$

Let $\mathbb{C}(t)$ be the field of rational functions $f(t) = \frac{p(t)}{q(t)}$ with $p,q\in\mathbb{C}[t]$. For instance, the function $g(t) = \sqrt{t}$ does not belong to $\mathbb{C}(t)$ but is ...
3 votes
0 answers
237 views

Cohomology of finite birational morphism

Let $f:Y \to X$ be a finite birational morphism between projective varieties (over $\mathbb{C}$) with $Y$ non-singular. I want to understand the cokernel of the pull-back morphism from $H^q(X,\mathbb{...
0 votes
1 answer
170 views

Moving general fibers of a fibration

Let $X$ be an irreducible projective variety over $\mathbb{C}$ admitting a morphism $\pi:X\rightarrow \mathbb{P}^1$ with connected fibers. We may assume that the general fiber of $\pi$ is smooth. My ...
1 vote
0 answers
109 views

Mori fiber space in dimension $2$ over a point is $\mathbf P^2$

Let $k$ be an algebraically closed field. Let $X$ be a surface over $k$. Let $\pi: X \to S$ be an extremal contraction. It is well-known that if $\dim(S) = 0$, then $X \cong \mathbf P^2$. I wonder if ...
3 votes
1 answer
342 views

Irreducibility of the base and of the general fiber

Let $f:X\rightarrow Y$ be a morphism of scheme over $\mathbb{C}$. Assume that $Y$ and the the general fiber $F_y = f^{-1}(y)$ of $f$ are irreducible. Does there exists an irreducible component $X'$ of ...
4 votes
1 answer
247 views

Question about valuation and blow up (a lemma in GIT book)

I'm reading Mumford's book Geometric Invariant Theory and confused about the proof of a lemma on Page 91&92: Lemma: Let $V_0$ be a smooth surface over an algebraically closed field $k$ with char$...
3 votes
1 answer
244 views

Moduli spaces and conic bundles

The moduli space $A_2(1,8)^{\operatorname{lev}}$ of $(1,8)$-polarized abelian surfaces with canonical level structure has a structure of conic bundle over $\mathbb{P}^2$ with a curve of degree $4$ as ...
9 votes
1 answer
381 views

Set theoretic equation for Veronese varieties

Consider the embedding $f:\mathbb{P}^n\rightarrow\mathbb{P}^N$ induced by the complete linear system of degree $d$ hypersurfaces of $\mathbb{P}^n$. Its image $V_{n,\,d}$ is degree $d$ Veronese variety ...

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