User moon - MathOverflow

Efficient algorithm finding 'a' solution of system of linear inequalities

2013-02-11T22:51:42Z

I'm working on rational number field $\mathbb{Q}$.

Is there an efficient algorithm finding a solution of system of linear inequalities?

In many computer algebra systems like Sage or Maple, there are functions finding the whole solution set, but in my problem (approximately 40 dimensional vector space with 600 inequalities) it seems that the computation is too heavy. Also, in my situation I don't need the whole set - just a single solution is sufficient. What is a good method to find a solution?

Answer by Moon for blow up of segre primal and $\mathcal{M}_{0,6}$

2013-01-28T04:35:26Z

As Steven said, there is a morphism $\overline M_{0,6} \to \tilde{X}$ because the inverse image sheaf of the ideal of double points generates a Cartier divisor. Now $\tilde{X}$ is nonsingular, so to check the fact $\overline{M}_{0,6} \to \tilde{X}$ is an isomorphism it suffices to show injectivity of the map, or show the equality of Picard numbers.

In general, the birational morphism $\overline M_{0,n} \to (\mathbb{P}^1)^n//SL_2$ is a composition of smooth blow-ups and Kirwan's desingularization. Consult http://arxiv.org/abs/1002.2461.

Example of rational projective variety of Picard number 1

2012-01-05T16:44:29Z

Is there any example (or more ambitiously, classification) of $X$ with following properties?

$X$ is a variety over $\mathbb{C}$;
$X$ is projective and normal;
$\rho(X) = 1$;
$X$ is birational to $\mathbb{P}^n$.

Also, I want to hear a result after adding a singularity condition: How about when $X$ is $\mathbb{Q}$-factorial? How about $X$ is non-singular?

I can't find an example which is not isomorphic to $\mathbb{P}^n$. The only result I know in this direction is Mori's theorem: If a nonsingular variety $X$ has ample tangent bundle, then $X$ is isomorphic to $\mathbb{P}^n$.

Checking normality of variety

2011-04-05T14:00:15Z

All varieties here is defined over complex number. Let $X$ be a normal projective irreducible variety and let $Y$ be a projective irreducible variety. Suppose that there is a bijective morphism $f : X \to Y$. This does not imply that $Y$ is normal. One can easily construct a counterexample for curve case.

My question is:

1) Is it true if $Y$ is regular in codimension 1?

2) Is there any extra condition guarantee normality of $Y$?

I searched some related questions and answers such as

http://mathoverflow.net/questions/12688/nonsingular-normal-schemes/12689#12689 or

http://mathoverflow.net/questions/60097/checking-whether-a-variety-is-normal ,

but I cannot find or prove a method working on my situation. In my problem, finding local defining equations is almost impossible.

Edit: One of motivation is a kind of inverse problem of Zariski's main theorem. Suppose that $g : X \to Y$ is a morphism between projective varieties with connected fiber from normal variety $X$. By taking Stein factorization, we can reduce to the situation of question.

Technique to prove basepoint-freeness

2010-09-03T02:16:21Z

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 I know is using Kawamata basepoint-free theorem:

Theorem. Let $(X, \Delta)$ be a proper klt pair with $\Delta$ effective. Let $D$ be a nef Cartier divisor such that $aD-K_X-\Delta$ is nef and big for some $a > 0$. Then $|bD|$ has no basepoints for all $b >> 0$.

Question. What other kinds of techniques to prove semi-ampleness or basepoint-freeness of given line bundle are?

Maybe I miss some obvious method. Please don't hesitate adding answer although you think your idea on the top of your head is elementary.

Addition : In my situation, $X$ is a moduli space $\overline{M}_{0,n}$. In this case, Kodaira dimension is $-\infty$. More generally, I want to think genus 0 Kontsevich moduli space of stable maps to projective space, too. $L$ is given by a linear combination of boundary divisors. It is well-known that boundary divisors are normal crossing, and we know many curves on the space such that we can calculate intersection numbers with boundary divisors explicitely.

Torsion-freeness of Picard group

2010-07-21T09:04:22Z

Let $X$ be a complex normal projective variety.

Is there any sufficient condition to guarantee the torsion-freeness of Picard group of $X$?

One technique I sometimes use is following: If $X$ can be represented by GIT quotient $Y//G$ for some projective variety $Y$ with well-known Picard group, then by using Kempf's descent lemma, we can attack the computation of integral Picard group.

Of course, if we can make a sequence of smooth blow-ups/downs between $X$ and $X'$ with well known Picard group, then we can get the information of $\mathrm{Pic}(X)$ from $\mathrm{Pic}(X')$.

Is there any way to attack this problem?

Dualizing sheaf of reducible variety?

2010-03-17T03:07:16Z

Sorry for my poor English.

Let $X$ be a reducible projective variety.

My question is:

How can I compute the dualizing sheaf of $X$ and express it in an explicit way?
Is there a method to get dualizing sheaf of whole reducible variety $X$ from the information of dualizing sheaves of its irreducible components?

Currently I'm not concern the general case, but I want a few accessible concrete examples such as:

reducible hypersurfaces,
union of toric varieties glued at isomorphic orbits(Alexeev calls it stable toric variety).

The reason why I concern is to understand the limit in moduli spaces of stable pairs.

Answer by Moon for Cohomology rings of GL_n(C), SL_n(C)

2010-03-19T03:34:11Z

For classical groups such as $GL_n(F)$, $SL_n(F)$ for $F = \mathbb{R}, \mathbb{C}$, $SU(n)$, $U(n)$ and $O(n)$, you may find the cohomology ring structure and its proof in M. Mimura and H. Toda, Topology of Lie groups I, translations of mathematical monographs, vol 91.

Comment by Moon

2012-01-05T21:45:03Z

Oh, I thought it too difficult way... Thank you for comments!

Comment by Moon

2011-04-08T06:00:35Z

Thank you. I'll try it.

Comment by Moon

2011-04-07T02:16:44Z

Dear Karl, thank you for great answer. I have another question. To show the normality of given variety, is it the best way checking R1 and S2? In my situation, given variety $X$ is an incident variety of points and curves in projective space. By dimension estimate, R1 is not too difficult, but I can't figure out how can I prove S2 (or CM) condition.

Comment by Moon

2010-11-05T01:43:18Z

To Michael Thaddeus, Thank you for a nice answer and comment, but I can't imagine that cone with finitely many extremal rays may not be polyhedral. Is there a counterexample or am I missing some obvious picture?

Comment by Moon

2010-09-04T10:49:34Z

Anyway, thank you for answer. By the way, what is the result of Mourougane and Russo? Do you mean the theorem 2.3.9 in PAG?

Comment by Moon

2010-09-04T10:46:02Z

You are right, nefness + bigness does not guarantee semi-ampleness. My question is following: Is there any sufficient condition to get semi-ampleness? With which conditions we get semi-ample property?

Comment by Moon

2010-09-04T06:43:59Z

I edited the question, thanks.

Comment by Moon

2010-08-31T07:01:26Z

Farkas proved that $\overline{M}_g$ is of general type for $g = 22$.

Comment by Moon

2010-04-12T01:58:20Z

Thank you! I'll check the paper.

Comment by Moon

2010-04-02T10:15:59Z

Red book(old print), 403p.