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2
votes
0answers
146 views

Is there an example of index 1 picardnumber 1 Fano 4-fold with h^{21}\neq0

Are there any known examples of index 1 smooth Fano 4-folds X with $\mathsf{Pic}(X)\cong\mathbb{Z}$ with $h^2(\Omega^1_X)\neq0$?
4
votes
1answer
317 views

Is the strict transform of a smooth subvariety under the iterated blowup of a smooth varitiety still smooth?

Suppose $Y$ is a smooth variety, $X$ is a smooth hypersurface of $Y$, and $D_1, \ldots, D_n$ are smooth subvarieties of $Y$ that intersect transversely such that the sum of the codimensions of the ...
8
votes
3answers
450 views

Birational automorphisms of canonical models

Let $X$ be a variety with canonical singularities such that $K_X$ is ample. Do you have a reference of the fact that every birational map from $X$ to itself is biregular? Thank you
7
votes
1answer
387 views

Cones, monoids, and the space of (very) ample divisors

An interesting and useful tool to study a projective variety is its ample cone. Understanding the structure of this cone reveals information about the variety, and it is an isomorphism-invariant so ...
2
votes
2answers
645 views

Rational map to a non-uniruled manifold

Let X, Y be smooth projective (over IC), let f:X..>Y be a rational map. Assume Y is not uniruled. Is it true that f will be regular over a non-empty open subset of Y? Funny..
18
votes
1answer
1k views

Rationality of intersection of quadrics

Let $X \subset \mathbb{P}^n$ be a complete intersection of two quadrics. It is classical that, if $X$ contains a line, then it is rational. The proof is very simple and basically it is given by taking ...
3
votes
1answer
452 views

Stable base loci

Let $\mathbb{B}(L):=\bigcap_{m \in \mathbb{N}} Bs(|mL|)$ be the stable locus of Cartier divisor. I have read the paper "Restricted volumes and base loci of linear sistems" in which it's proved that ...
5
votes
0answers
499 views

Semistable sheaves on rationally connected manifolds

Let $(X,H)$ be a polarized projective manifold of dimension $n$ defined over $\mathbb C$, and let $\mathcal E$ be a reflexive sheaf on $X$. If for every subsheaf $\mathcal F \subset \mathcal E$ the ...
5
votes
3answers
538 views

When does f-nef imply nef (after twisting?)

For a surjective morphism $f \colon X \to Z$ of smooth projective varieties, a line bundle $L$ on $X$ is $f$-nef if is nef on the fibers of $f$. More precisely, for every curve $C$ that is mapped to a ...
15
votes
3answers
707 views

What is known about the birational involutions of P^3?

Describing the group of birational automorphisms of $\mathbb{P}^n$, $\mbox{Bir}(\mathbb{P}^n)$, for $n\ge 3$ is a fundamental open problem in birational geometry. For $n=2$, the classical theorem of ...
3
votes
1answer
359 views

Positivity of the anticanonical bundle of a rationally connected manifold

Let $X$ be a rationally connected smooth projective variety defined over $\mathbb C$. (1) Can we find a surface $S \subset X$ such that $ (-K_X)^2 \cdot S > 0 ? $ If yes, can we assume that $S$ ...
7
votes
1answer
407 views

Simplified treatment of resolutions of complex analytic varieties?

According to the article of Hauser: The Hironaka theorem on resolution of singularities http://www.ams.org/journals/bull/2003-40-03/S0273-0979-03-00982-0/home.html The existence of resolution of ...
5
votes
2answers
511 views

Topology of the preimage of a point for degree one holomorphic maps

Let $M^n$ and $N^n$ be two compact complex (or complex projective) manifolds. Let $f: M\to N$ be a holomorphic map of degree one. How to prove that for each $x\in N$ the set $f^{-1}(x)$ is ...
1
vote
0answers
315 views

birational equivalence and mirror CYs

If a CY X is a mirror to Y then any CY Z which is birational to X is also a Mirror of Y. This is the motivation for the Kawamata's "moveable Kahler cone" which includes the Kahler cones of all the CYs ...
1
vote
1answer
350 views

Numerically rigid nef divisor

Is it possible to find an example of an $\mathbb{R}$-Cartier divisor $D$ on an irreducible variety $X$ that is non-trivial, nef, effective and numerically rigid? By "numerically rigid" I mean that ...
10
votes
0answers
562 views

Relative canonical divisors

Suppose that $X$ is a Gorenstein variety and that $\pi : Y \to X$ is a birational map of varieties with normal $Y$. In this case the relative canonical divisor is defined to be $K_Y - \pi^*K_X$ (if ...
9
votes
2answers
797 views

Blowups of Cohen-Macaulay varieties

Suppose that $X$ is a Cohen-Macaulay normal scheme/variety and $\pi : Y \to X$ is a proper birational map with $Y$ normal. Question: Is $Y$ also Cohen-Macaulay? Are there common conditions which ...
3
votes
2answers
1k views

Technique to prove basepoint-freeness

Let $X$ be a smooth projective variety over $\mathbb{C}$. And let $L$ be a big and nef line bundle on $X$. I want to prove $L$ is semi-ample($L^m$ is basepoint-free for some $m > 0$). The only way ...
7
votes
1answer
727 views

Appropriate journal to publish a determinantal inequality

I have recently made the following observation: Let $v_i := (v_{i1}, v_{i2})$, $1 \leq i \leq k$, be non-zero positive elements of $\mathbb{Q}^2$ such that no two of them are proportional. Let ...
3
votes
2answers
571 views

Possible singularities of the base of a Mori fiber space

Suppose X is a normal projective complex variety, (X, $\Delta$) is a klt pair and f : X $\to$ Z is a Mori fiber space given by a contraction of an extremal ray for this pair. Here I mean that the ...
17
votes
3answers
2k views

Birational invariants and fundamental groups

In pondering this MO question and people's efforts to answer it, and recalling also something that I learned in my youth about using Morse theory ideas to prove some results of Lefschetz in the ...
31
votes
5answers
2k views

Surfaces in $\mathbb{P}^3$ with isolated singularities

It is classically known that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only ordinary singularities, i.e. a curve $C$ of double points, ...
7
votes
2answers
710 views

How much can small modifications change the nef cone?

First let me give a precise formulation of the question; I'll give some background/motivation at the end. If X is a projective variety which is Q-factorial (meaning X is normal, and some sufficiently ...
3
votes
1answer
330 views

A question about the construction of Francia flip

Here is the construction. Start with $U$ a normal variety of dimension 3 with a unique singular point locally isomorphic to the quotient of $(x, y, z) \to (-x, -y, -z)$. Also assume we have a small ...
9
votes
1answer
406 views

Schemes as a model category

I'm just learning some basics of model categories, so please forgive me if my question turns out to be trivial. I hope it does at least make sense. A natural temptation is to relate this machinery to ...
3
votes
1answer
578 views

Birational pullbacks of divisors on singular varieties

Actually I have two related questions. Here is the first... Suppose $X$ is a, possibly singular, complex projective variety. Let $D$ be an effective Cartier divisor on $X$ and $x\in X$ a closed ...
2
votes
1answer
355 views

About b-divisors

In the last period I have studied a number of papers of O. Fujino and F. Ambro using the language of b-divisors. So far it seems to me that every proof I have studied can be translated in the ...
7
votes
4answers
1k views

Is the Euler characteristic a birational invariant

Suppose that $X$ and $Y$ are smooth projective varieties which are birationally equivalent. I would like to have that $$\textrm{deg} \ \textrm{td}(X) = \textrm{deg} \ \textrm{td}(Y).$$ Invoking the ...
2
votes
3answers
938 views

The Riemann correspondence for riemann surfaces made explicit and its generalizations

Riemann showed (not proved rigorously) that there is a correspondence between compact riemann surfaces and algebraic function fields in one variable (does anyone know the year?). To construct the ...
7
votes
3answers
1k views

Contracting divisors to a point

This is quite possibly a stupid question, but it is pretty far from what I normally do, so I wouldn't even know where to look it up. If $X$ is a projective variety over an algebraically closed field ...
7
votes
0answers
184 views

Projectivity of flops

Say I have a projective (smooth, compact) irreducible symplectic variety $X$ over $\mathbb{C}$ and I perform a Mukai flop. It is well known that if the resulting variety $\widetilde{X}$ is Kahler, it ...
7
votes
1answer
497 views

Are groups in (Var/k, rational maps) necessarily algebraic groups?

Let $RVar$ be the category which has complex algebraic verieties as objects and rational maps as morphisms [Edit: for $RVar$ to be a category, the rational maps have to be dominant]. Let's consider a ...
9
votes
2answers
468 views

Minimal model which is necessarily singular

I was told during a summer school on the MMP a nice example (which I have also mentioned here on MO) that I'm not able to figure out anymore. The example (due, I think, to Miles Reid) is a smooth ...
1
vote
2answers
367 views

Relative minimality for conic bundles

Due to difficulties in finding references I am not sure of what relative minimality is about for conic bundles. Suppose we're working on con. bun. over surfaces. The definition is ok: the fiber over ...
7
votes
1answer
415 views

Does a birational involution of C^n always have a fixed point?

The title describes completely the question. For n=1 it is an easy exercise. For n=2 the statement is still true, and depends on the classification of birational involutions of P^2 (see e.g. ...
10
votes
2answers
594 views

Rationality of GIT quotients

I recently worked through most of the proof of the rationality of the moduli of genus 3 curves, which seemed to have the following structure: Every nonhyperelliptic genus 3 curve is a smooth plane ...
3
votes
3answers
278 views

Nefness of $h-e$ in the blowup of $\mathbb{P}^n$

Let $S$ be the blow up of $\mathbb{P}^n$ in a point $P$. Let $h$ be the class of the pullback of an hyperplane of $\mathbb{P}^n$ and $e$ the class of the exceptional divisor. Why is the divisor ...
1
vote
5answers
478 views

Reference for explicit calculation of blowups (of ideals) and strict transforms

Can one suggest some references where explicit calculations for blow up technique(along an ideal) and strict transformation is done in different examples?
6
votes
3answers
414 views

Why is the Hodge class of \bar{M_g} big and nef?

Let pi: \bar{Mg,1} \to \bar{M_g} be natural projection of compactified moduli stacks of curves and let omega be the relative dualizing sheaf. Then the Hodge class \lambda of \bar{M_g} is the first ...