User jesus martinez garcia - MathOverflow

Minimal resolution of Log del Pezzo surfaces

2011-08-04T12:52:21Z

Suppose $X$ is a log del pezzo projective surface of index $l$. As far as I understand it will have a finite number of singular points all of which can be resolved by sucessive blow-ups.

Let $E_i$ be the exceptional divisors of the minimal resolution. Their self-intersection numbers are $E_i^2\leq -2$. Is there a lower bound on these numbers?

Intuitive pictures in characteristic p

2012-12-20T10:10:13Z

This is a tough one, but does anyone know of any images that recall characteristic p geometry (over algebraically closed fields) in some sense? It is not enough if it is some picture that can be also understood solely in characteristic 0.

A quick search through the literature has proved fruitless.

I have been thinking for a while of asking this question, but I never had a pressing need rather than my own curiosity. However, now I am trying to improve a poster by including pictures but the topic is algebraic geometry in characteristic p.

An example of such an image in Complex Geometry would be the arrangement of contracted and blow up curves in the standard Cremona transformation of the projective plane. The one in this poster is simple but effective.

I believe this question also might be of interest for people who try to explain research to non-mathematicians or simply to mathematicians who are not geometers.

Bertini's Theorem small print

2012-10-11T17:58:02Z

Suppose $S\subset \mathbb{P}^n$ is a smooth del Pezzo surface and $C$ is an irreducible smooth curve (you can make it rational if it simplifies the setting) such that $\mathcal{L}=\vert -K_S-C\vert $ is non-empty. To simplify you may consider $-K_S$ is very ample and let me deal with the degree $1$ and $2$ cases. Moreover suppose that $h^0(S,\mathcal{O}(\mathcal{L}))\geq 2$ (i.e. $\mathcal{L}$ is at least a pencil).

It was my understanding that by Bertini's theorem one could choose a general member $L\in\mathcal{L}$ such that $L$ is smooth (and reduced and connected). I have been told this is wrong and after going to Hartshorne (and Wikipedia and some expository paper by Kleiman that Francesco added to the comments) I am also of the opinion that it may actually be wrong, but that $L$ must be irreducible away of the base locus of $\mathcal{L}$.

However I am unable of providing a proof nor a counter-example. Does someone have an insight on this? I also suspect the base locus of $\mathcal{L}$ may actually be empty.

Edit: Originally $H$ was a hyperplane section. The question is actually motivated by 'the' hyperplane section so I have rephrased it to meet this point. Apologies for the confusion.

Applications of Slope Stability

2012-10-15T13:27:36Z

Ross and Thomas developed slope-stability of $(X,L)$ where $X$ is an $L$-polarised variety and $L$ is an ample line bundle, as an obstruction to K-stability of $(X,L)$.

DISCLAIMER: (Forgive me if I don't define what these are, but for the purpose of the question if you do not know them well already is not going to help you. Moreover the definition is rather technical. Also, I am not an expert in these topics, so I expect to say a couple of things wrong without being aware of it)

K-stability is an obstruction to the existence of Kahler-Einstein metrics on $X$. Therefore slope-stability can be seen as a tool to decide when a Fano variety does not admit a Kahler-Einstein metric:

Kahler einstein $\Rightarrow$ K-stable $\Rightarrow$ slope-stable.

The last arrow is not strict, i.e. there are slope-stable $(X,L)$ which are not K-stable. On the other hand, computing slope-stability is much easier than computing K-stability.

I know that K-stability has other applications, for instance if $(X,\mathcal{O}(-mK_X)),\ m\in \mathbb{Z}_{>0}$ satisfies certain conditions (including being Fano) and K-semistability, then the singularities of $X$ are log terminal by a result by Odaka in Annals. Therefore K-stability is interesting not only within the Kahler-Einstein problem.

I was wondering if there is any other applications of slope-stability other than as an obstruction to K-stability.

(answers considering the log setting are also welcomed)

The canonical line bundle of a normal variety

2010-08-16T09:06:18Z

I have heard that the canonical divisor can be defined on a normal variety X since the smooth locus has codimension 2. Then, I have heard as well that for ANY algebraic variety such that the canonical bundle is defined:

$$\mathcal{K}=\mathcal{O}_{X,-\sum D_i}$$

where the $D_i$ are representatives of all divisors in the Class Group.

I want to prove that formula or I want to find a reference for that formula, or I want someone to rephrase it in a similar way if they heard about it.

Why do I want to prove it? Well, I use the definition that something is Calabi Yau if its canonical bundle is 0. In the case of toric varieties, $\sum D_i$~0 if all the primitive generators for the divisors lie on a hyperplane. Then the sum is 0 and therefore the toric variety is Calabi-Yau.

Can someone confirm or fix the above formula? I do not ask for a debate on when something is Calabi-Yau, I handle that OK, I just ask whether the above formula is correct. A reference would be enough. I have little access to references at the moment.

Example of cone of numerically effective curves which is not polyhedral

2012-01-19T18:17:14Z

I think I have seen more than one reference in which the cone of numerically effective curves can be 'not polyhedral', i.e. with an infinite number of extremal rays

I cannot remember where I read that the usual example is the blow-up of the plane in the $9$ points of intersection of $2$ general cubics. This should give an infinite number of $-1$-curves, but I don't manage to see why it should be infinite! As far as I see the strict transform $C$ of any curve in the pencil would give me $C^2=\pi*(C)+ \sum E_i^2=9-9=0$ where $E_i$ are the exceptional divisors.

What am I missing?

Controlling singularities on log mmp

2012-04-01T15:43:35Z

Suppose all my varieties are complex threefolds $X\rightarrow Y$ over some smooth base curve germ $Y$. We can assume the fibres are Del Pezzo surfaces with generic smooth fibre.

If I do (relative) log mmp over a klt pair $(X,D)$ what I obtain is a klt pair $(X',D')$ as output.

If I do (relative) mmp over a terminal $X$, I obtain a terminal $X'$ as output.

Now suppose $(X,D)$ is terminal. I would like to apply (relative) log mmp over it to obtain $(X',D')$ terminal as output. Of course this will not happen in general, but I was wondering if there are:

1) sufficient conditions on $(X,D)$ for this to happen, or

2) whether there is/may be some freedom when running MMP to make a choice in some/all the steps to guarantee this.

Probably the answer to both questions is NO, or even worse: 'probably not. I thought I would give it a try, anyway. It is kind of too specific to find it out of a book without learning all MMP (which I am actually trying to do at the same time).

'Reference' request: Program to work with cyclic quotient singularities.

2012-02-25T22:48:32Z

I'm looking for program code to deal with cyclic quotient singularities on normal surfaces. In particular, at the moment I need that given a singularity $p$ like $p=\frac{1}{n}(1,a)$ the code computes:

Its continuous fraction decomposition (i.e. $\frac{1}{n}(1,a)=[b_1,\ldots,b_k]$). Of course, the $b_i$ are the self-intersection numbers of the exceptional curves of the minimal resolution.
The discrepancies of the resolution, i.e. if the minimal resolution of $p$ is $f:Y\rightarrow X$ and has exceptional curves $E_i$, then we can write $$f^*(K_X)=K_Y+\sum a_i E_i.$$ I want to find the values of $a_i$ automatically.

I know there is code written for 1. for instance in Magma or in several pages on the Internet, but I haven't seen any for the second one, and having asked around no one seem to know about it.

I know both things are pretty much automatic, for the second one we just need to intersect the above formula with each E_i, use the genus formula and resolve a linear system of equations. It should not be a difficult task (albeit a boring one) to write this code myself in Maple (which is the only mathematical language I've ever used) but I'd like to save myself the hassle and time. Moreover, I will most likely need to add code for other computations on top of this code. Code that maybe (let me dream), other people may useful in the future. I'd rather prefer to add something to an existing library than have several partial-functional libraries in different languages around.

Also, if no one knows about (2) but knows code or settings in Magma, Macaulay2... that deals easily with (1), please refer me to it and I'll build (2) over that.

Thanks!

Elementary short exact sequence of sheaves

2011-04-26T18:23:47Z

This question arised when I was trying to use this answer to understand Reid's "Young Person's guide to Canonical Singularities". In particular page 352 when computing the blow-up $Y\rightarrow A^2/\mu_3$, the affine plane quotient the cyclic group of order 3, arises to the conclusion that the exceptional divisor is $E\sim P^1$, (no problems there) and $\mathcal{O}_E(-E)\sim \mathcal{O}(3)$ (problems here).

Given a variety $Y$ and an effective Cartier divisor $D$ on it, there seems to be a pretty standard exact sequence:

$$0 \longrightarrow \mathcal{O}_Y \longrightarrow \mathcal{O}_Y(D) \longrightarrow\mathcal{O}_D(D)\longrightarrow 0 $$ As far as I understand, if $U$ is an open set in $S$ and $D\cap U = div(g)_U$ (for $D$ a hypersurface, if you want, and extend by linearity), then

$$ \mathcal{O}_Y(D)(U)= \{g \in \mathcal{O}_Y(U) \vert div(g)\geq D \}$$

or equivalently $g/f$ is regular. The first map must be something like $g\rightarrow gf$ maybe with some order. A good answer to my question would include:

Is this correct?
What is $\mathcal{O}_D(D)$?
What is the second map?
What does $\mathcal{O}_D(-D)$ mean
Why $\mathcal{O}_E(-E) \sim \mathcal{O}(3)$? I understood the RHS is generated by polynomials of degree 3?

I am aware this is a simple question and probably everyone knows why, but I could not find a proper answer for it.

Numerically negative exceptional divisor on a surface.

2012-01-19T18:09:32Z

Suppose $S$ is an algebraic surface (possibly projective) over an algebraically closed field $k$. Suppose $D_i$ are irreducible smooth curves (rational, if you want) with negative self-intersection and simple normal crossings such that there is a morphism $f:S\rightarrow S'$ which collapses them to a point.

Can we form an effective divisor $D=\sum d_i D_i$ such that $D\cdot D_i<0$ for all $i$? Note that if I had said $\leq$ rather than $\lt$ this is the content of the Proposition in page 83 of Reid's Park City lectures. I ask for strict inequality.

My belief is that this is possible, but I don't manage to give a proof nor find a reference or even to find a counter-example... This would be very useful for many arguments in birational geometry of surfaces.

Minimal Model Program for surfaces over algebraically closed fields of characteristic p

2012-01-11T17:07:51Z

Let $k$ be an algebraically closed field of characteristic $p>0$.

I have been trying to find out unsuccessfully if there is a mmp for algebraic surfaces over $k$. I know minimal surfaces are classified thanks to Zariski, Mumford, others in this setting and that is not my question. I want to 'run' LMMP of pairs not to find minimal model but as a tool to prove several stuff.

In particular I would like to know if there is a Cone Theorem (i.e. giving me that extremal rays have non-positive self-intersection, I suspect the answer is yes) and if there is a contraction theorem (I suspect not yet).

By contraction theorem I mean that if given $(X,D)$ where $X$ is a surface $D$ is an effective divisor and the pair is klt, if $K_X+D$ is not nef I can find a curve $E$ with negative self-intersection such that $E$ can be contracted.

I am aware I am being vague with the formulation but I do not want to constrain your imagination.

Now, if someone also knows if flips and termination of flips are possible, please share :)

Answer by Jesus Martinez Garcia for How can we find a surface with a given singularity?

2012-01-13T17:12:59Z

I'm asumming you assume ground field $\mathbb{C}$. I actually wondered about the same thing a while ago, in the case of surfaces. I find hard to think in particular embeddings a priori and then the singularities on it but I'd rather think first on the singularities and then think where they can be embedded in. You speak of $A_n$ which in the case of surfaces is a Du Val singularity and it is given by certain equations anallytically which you can find in Reid's notes. However if you start combining several singularities in the same surface and for instance fix the degree, it is intuitive that you cannot glue the different analytic patches together. It is therefore 'easier' to start with a surface with given singularities, degree, Picard number... and check whether it exists or not.

A classification of log Del Pezzo surfaces of index $\leq 2$ was done by Alexeev and Nikulin. I think more general cases are unknown. Higher dimensional cases are probably even more complicated. This classification has been used for instance, to classify certain 3-folds with $T$-singularities (see Hacking-Prokhorov)

This is not a great answer, but maybe it helps to point you on references to read upon. A particular example may be easy, but a general picture I am afraid that requires a long road for which I have not read the map completely.

Graph of dependencies from a Latex file

2012-01-05T17:12:44Z

This question has been "manually migrated" to TeX-SX: http://tex.stackexchange.com/q/40200/86

Apologies if the question is not very appropiate for Mathoverflow. It seems to me more appropiate here than in the other 'exchange' sites.

Is there an IT tool to create a graph of dependencies from a Latex file? The sense is the following:

It just occurred to me that if everyone creates propositions with proofs (usually) afterwards and these proofs use \eqref, \ref \cite to call to other results it should be feasible to create a graph of dependencies of results, given a paper written in Latex.

I think such a thing would be useful for any mathematician (check dependencies, recursive arguments, that there are lemmas which have a need, possibility of suggesting equivalences, writing well-ordered documents...) so I would be surprised if this does not exist yet, but I couldn't find it anywhere.

Log resolutions on surfaces and 3-folds in characteristic p

2011-10-24T14:03:17Z

If $X$ is a normal projective variety and $D$ a divisor in it, we say that $\pi\colon (\widetilde X,\widetilde D)\rightarrow (X,D)$ is a log resolution if $\widetilde X$ is a resolution of $X$, the strict transfor $\widetilde D$ of $D$ is non-singular and $\widetilde D \cup { \text{exceptional divisors} }$ have simple normal crossings.

I know little about techniques for resolution of singularities and as far as I am aware, for varieties over algebraically closed fields, the problem of finding resolutions of singularities is open.

However, I was wondering if the following is solved, by whom and if someone can provide me with a 'black-box' reference:

Question: Given a projective variety $X$ over a field of characteristic $p$ and a divisor $D$ on $X$, is there a log resolution of the pair $(X,D)$ in the cases where $X$ is a non-singular variety of small dimension (1,2,3) and/or in the case the $p\neq 2,3,5\ldots$? What if the variety is a product of a projective variety and the affine line?

Of course partial answers are appreciated. However the purpose of this is just to use it in a birational proof for something else, so by no means I intend to prove it myself or get any close to it.

Del pezzo surfaces in positive characteristic

2011-09-22T18:04:18Z

For me a Del Pezzo surface $X$ over an algebraically closed field of characteristic $p$ is an algebraic surface where the anticanonical bundle $\omega^{-1}_X$ or $-K_X$ is ample. (I prefer the second notation, although it is not very correct),

In characteristic 0, as far as I know, there is a classification. $X$ has to be isomporphic either to $\mathbb{P}^1\times \mathbb{P}^1$ or a blow-up of $\mathbb{P}^2$ at $9-K_X^2\geq 0$ points in general position, i.e. not 3 points in the same line and not 6 points in the same conic of $\mathbb{P}^2$.

I would like to know if there is such a classification in characteristic $p\geq 2$. As far as I know the classification is definitely the same for $p>3$, and probably even for $p=3$ and there are 'extra' surfaces for $p=2$.

The perfect answer would confirm whether these assertions are true, describe the extra cases (in particular in terms of which curves can live in them or minimality) and/or give a reference.

But anything is better than nothing, so even if you know a bit of this I would like to know.

Generalisations of Riemann-Roch for surfaces

2011-08-08T13:45:27Z

Let $X$ be a smooth projective algebraic surface (over $\mathbb{C}$ ). For all $L\in \mathrm{Pic}(X)$, we have $$\chi(L)=\chi(\mathcal{O}_X)+\frac{1}{2}(L^2-L\cdot \omega_X).$$ This is the famous Riemann-Roch theorem in the flavour I like the most. It usually comes together with the following two formulas: $$\chi(\mathcal{O}_S)=\frac{1}{12}(K_X^2+\chi_{top}(X)),$$ the Noether's formula and $$2p_a(C)-2=C^2+C\cdot K_X,$$ the genus formula for $C$ an irreducible (possibly singular) curve.

Is there a similar (or maybe the same) version for a) Smooth quasi-projective surfaces. b) Projective surface with quotient singularities, or A-D-E singularities.

Answer by Jesus Martinez Garcia for How to resolve a disagreement about a mathematical proof?

2011-08-04T14:06:51Z

I would add something else. Talk to someone who has used this result before. Hopefully there will be someone and he/she will have read the proof. I think it is easy to find who has referenced a paper through mathscinet. Obviously this person should be accessible to you, but it worths a try (it could be someone you have worked with and you feel more comfortable with).

And keep being tactful :)

Global sections of a linear system

2011-07-12T11:31:24Z

Recently, following Beauville's book (exercises iv.(1),(2)) I have been working on Hirzebruch surfaces (from the algebraic geometry point of view) and I had to compute the space of global sections of several linear systems. For instance, if $h$ is a section of $\mathbb{P}^1$ and $f$ is a fibre, then I would be looking for $h^0(\mathbb{F}_n,\vert h+kf\vert),\ k\geq 0$. The reason to do this was to find morphisms into projective space.

Also I was interested in knowing the usual stuff: whether the linear system had fixed points, whether it separated points and tangents, in which points it didn't...

My approach was to do it for $\mathbb{F}_0\cong \mathbb{P}^1\times \mathbb{P}^1$ and then by induction find it for the rest of $\mathbb{F}_n$ using elementary transformations.

It seemed to me a bit 'ad hoc' and I feel the only reason I could do this is because I had a very explicit knowledge of $P^1\times P^1$ and how to find the other surfaces from this one. If I had started with the image via that linear system into projective space, even with lots of information about it, I doubt I had been able to find so much information or even understand which curves were linearly equivalent.

Question: Are there methods to find information (dimension, base points, incidence) about linear systems of divisors in a surface given (some) explicit information about the geometry of that surface? By information I mean configuration of lines, degree, whether it has curves embedded inside, intersection of particular curves...

I am aware this is a bit of a vague question, but that is precisely the point, I do not seek solutions to particular examples but tools that work for as many surfaces as possible.

Also, I do not look for methods that apply to rational surfaces by looking at curves in the plane.

Answers which are general to higher dimensions are valuable too.

Which 'well-known' algebraic geometric results do not hold in characteristic 2?

2011-07-06T12:42:04Z

A smooth curve $X$ in $\mathbb{P}^n$ is strange if there is a point $p$ which lies on all the tangent lines of $X$.

Examples are $\mathbb{P}^1$ is strange and so is $y=x^2$ in characteristic $2$. These are in fact all (see Hartshorne IV.3.9).

For this reason, in any characteristic we do not use lines to get embeddings in projective space.

But this actually has consequences. Many classical theorems in algebraic geometry do not use 'transcendental methods', i.e. the only result they use is that the base field is algebraically closed, and so they can be applied in finite characteristic. Or they cannot?

Here is where characteristic $2$ breaks down the standard results. For instance, when embedding blow-ups of $\mathbb{P}^2$ in $\mathbb{P^n}$ we use linear systems of conics and cubics in $\mathbb{P}^2$ to separate the points, but this is not possible in characteristic $2$ (have a look at Beauville IV.4 if you do not know how linear systems can embed spaces). This means that in characteristic 2 we cannot interpret cubic surfaces ni $\mathbb{P}^3$ in terms of blow-ups of the projective plane in 6 points in general position and viceversa.

OK, enough intro. My question is: "Are there other examples of results which do not apply in characteristic 2 due to other reasons not involving embeddings in projective space?"

Also, I am looking for results that do not hold in low characteristic. i.e. I know that vanishing theorems do not hold in characteristic p in general, but I am looking for pathologies for some but not all finite characteristic (usually 2, or 3).

I suspect 'probably yes but not many', since I cannot come up with any, but if it turns out there are lots, maybe I'll make this question community wiki.

Elementary transformations of ruled surfaces as maps of vector bundles

2011-06-30T09:56:03Z

This comes as a question in Beauville's Algebraic surfaces book (III.24 (2)). We work over $\mathbb{C}$.

All geometrically ruled surfaces (grs) $p:S\longrightarrow C$ over a curve $C$ can be seen as $S=\mathbb{P}(E)$ where $E$ is a vector bundle of rank $2$ over $C$, i.e. a locally free sheaf of rank $2$ over $C$. They are always minimal, or relatively minimal, depending on your book.

An elementary transformation $S\dashrightarrow S'$ with centre $s\in S$ corresponds to blowing up $s$ and contracting the strict transform of a fibre. This gives another grs $S'$ over $C$.

That point of view which is explicit and algebraic is the one I understand and usually work with. But you can also think in the following way. Take $s\in S$ and consider the pushforward of the skyscraper sheaf $\mathbb{C}(s)$, $F=p_*\mathbb{C}(S)$. Since $s$ can be seen as $(p(s),D_s)$ where $D_s\in E_{p(s)}$ is a line at the stalk of $E$ at $p(s)$, we get a map $$u_s:E\longrightarrow F$$ which I understand it can be defined on the stalks and sends $(c,v)$ to $(c,0)$ if $c\neq p(s)$ and to $(c,v+D_s)$ otherwise. Here we identify $F_s$ with $E_s/(D_s)$ or some similar quotient. Beauville did not write $F$ as a pushforward but as the skyscraper sheaf itself. I think that is an unimportant typo. Define $E'=ker(u_s)$.

The question is:

1) $E'$ is a vector bundle of rank $2$.

2) $S'=\mathbb{P}(E')$ corresponds to the elementary transformation with centre $s$ $S\dashrightarrow S'$.

3) That transformation corresponds to the inclusion $E\rightarrow E'$.

I find hard to believe 1) is true although it must be because it is written in the book and an article by Hartshorne says it is 'easy' to see. I fail to see how the rank of $E'$ is going to be $2$ over $p(s)$. It seems to me that in the short exact sequence $$E'\longrightarrow E \longrightarrow F,$$ if you restrict to stalks you necessarily have $E'_s\cong \mathbb{C}$, but I must be being somewhat naive.

Probably the problem is that I do not understand $u_s$. I probably can do 2) and 3) once I solve 1). Any ideas?

Answer by Jesus Martinez Garcia for Best online mathematics videos?

2011-05-12T04:31:17Z

All the talks of Atiyah 80+

Answer by Jesus Martinez Garcia for Best online mathematics videos?

2011-05-12T04:10:58Z

David Cox's lectures in toric varieties at MSRI

Something really good to end the evening with :)

Answer by Jesus Martinez Garcia for Jokes in the sense of Littlewood: examples?

2011-05-12T04:21:30Z

The journal of unpublishable mathematics, which seems to be down at the moment is one of my favourites

Answer by Jesus Martinez Garcia for Mathematical ideas named after places

2011-05-12T04:05:45Z

Italian Algebraic Algebraic Geometry

One that is not but I used to think so: Catalan number :)

Are all parametrizations via polynomials algebraic varieties?

2010-08-31T08:24:52Z

Suppose that we have a parametrization via polynomials as follows:

$$t\longrightarrow (f_1(t),\ldots,f_n(t)),$$

where $t$ is a vector in $\mathbb{C}^r$ and $f_i$ are polynomials of arbitrary degree.

Can we always find equations such that the image is an affine algebraic variety?

The question is motivated by Exercise 1.11 in Hartshorne:

Let $Y\subseteq A^3$ be the curve given parametrically by $x = t^3, y= t^4, z = t^5$. Show that $I(Y)$ is a prime ideal of height 2 in $k[x,y,z]$ which cannot be generated by 2 elements.

I am not interested in the exercise in particular. Finding the variety is easy sometimes, for instance $t\rightarrow (t^2,t^3)$ is given by $I(x^3-y^2)$.

I am looking for a result which says that the image is always an affine algebraic variety AND a procedure to find the ideal.

Ringed and locally ringed spaces

2010-09-10T14:39:02Z

A pair $(X,O_X)$ is a ringed space if $X$ is a topological space and $O_X$ is a sheaf of rings. If every stalk $O_{X,x}$ is a local ring, then we say that $(X,O_X)$ is a locally ringed space.

In the case of $X$ being an abstract algebraic variety, not necessarily irreducible, and $O_X$ its sheaf of regular functions:

1) Is it possible to have a ringed space which is not a locally ringed one? And if $X$ is irreducible? My guess is NO for the first question and YES for the second.

Correct me if I am wrong. My reasoning goes as follows, since stalks are local we can work on the affine case. All irreducible algebraic varieties are quotient of $K[x_1,\ldots,x_n]$ by a prime ideal and prime ideals always have a unique maximal ideal, corresponding to a point, so the quotient on the point is a field. This seems to work for $K=\mathbb{C}$. In the irreducible case I am not sure of what happens, but for the complex case you can have two lines crossing and I get each of them may work independently giving two local ring ideals? Pure speculative...

2) Is this true for non-algebraically closed fields or fields with positive characteristic?

3) Could you please provide with any example of non-locally ringed spaces? In varieties or schemes, if it is possible.

4) If $X$ is a variety/scheme, is there any example of morphism of ringed spaces which are locally ringed but which is not a morphism of locally ringed spaces? (i.e. for $f\colon X\rightarrow Y$ there is a sheaf morphism $f^\sharp\colon O_Y \rightarrow f_* O_X$ which is not a morphism of local rings on the stalks).

These questions are a bit vague but hopefully you understand what I mean.

Answer by Jesus Martinez Garcia for Notable mathematics during World War II

2010-08-21T08:18:32Z

Grothendieck went to Vietnam to deliver lectures and a report of what he did can still be found online.

Bertrand Russell was imprisoned during WWI for anti-war activities and wrote "Introduction to Mathematical Philosophy" (1919) while in prison.

Hardy, in protest for Russell's affair, left Cambridge to Oxford and continued working there and collaborating by mail with Littlewood. Both of them worked during that time in Mathematics and there is fiction written about it.

Answer by Jesus Martinez Garcia for Counting/constructing Toric Varieties

2010-08-13T10:02:18Z

As far as my understanding goes the answer is no, and I will try to explain why and clarify the list of comments (I have little reputation so I cannot comment there) and give you a partial answer. I hope I do not patronise you, since you may now already part of it.

First of all, as Torsten said, it depends what you understand for classification. In this context a torus $T$ of dimension $r$ is always an algebraic variety isomorphic to $(\mathbb{C}^*)^r$ as a group. A complex algebraic variety $X$ of finite type is toric if there exists an embedding $\iota: (\mathbb{C}^\ast)^r \hookrightarrow X$, such that the image of $\iota$ is an open set whose Zariski closure is $X$ itself and the usual multiplication in $T=\iota((\mathbb{C}^\ast)^r)$ extends to $X$ (i.e. $T$ acts on $X$).

Think about all toric varieties. It is hard to find a complete classification, i.e. being able to give the coordinates ring for each affine patch and the morphisms among them for all toric varieties.

However, when the toric varieties we consider are normal there is a structure called the fan $\Sigma$ made out of cones. All cones live in $N_\mathbb{R}\cong N\otimes \mathbb{R}$ where $N\cong \mathbb{Z}$ is a lattice. A cone is generated by several vectors of the lattices (like a high school cone, really) and a fan is a union of cones which mainly have to satisfy that they do not overlap unless the overlap is a face of the cone (another cone of smaller dimension). There is a concept of morphism of fans and hence we can speak of fans 'up to isomorphism' (elements of $\mathbf{SL}(n,\mathbb{Z})$). Given a lattice N, there is an associated torus $T_N=N\otimes (\mathbb{C}^*)$, isomorphic to the standard torus.

Then we have a 1:1 correspondence between separated normal toric varieties $X$ (which contain the torus $T_N$ as a subset) up to isomorphism and fans in $N_\mathbb{R}$ up to isomorphism. There are algorithms to compute the fan from the variety and the variety from the fan and they are not difficult at all. You can easily learn them in chapter seven of the Mirror Symmetry book, available for free. Given any toric variety (even non-normal ones) we can compute its fan, but computing back the variety of this fan many not give us the original variety unless the original is normal. You can check this easily by computing the fan of a $\mathbf{V}(x^2-y^3)$ (torus embedding $(t^3,t^2)$) which is the same as $\mathbb{C}^1$ but obviously they are not isomorphic (the former has a singularity at (0,0)). In fact, since there are only two non-isomorphic fans of dimension 1 (the one generated by $1\in \mathbb{Z}$ and the one generated by 1 and -1) we see that there are only three normal toric varieties of dimension 1, the projective line and the affine line, and the standard torus.

The proof of this statement is not easy and to be honest I have never seen it written down complete (and I would appreciate any reference if someone saw it) but I know more or less the reason, as it is explained in the book about to be published by Cox, Little and Schenck (partly available) This theorem is part of my first year report which is due by the end of September, so if you want me to send you a copy when it is finished send me an e-mail.

So, yes, in the case of normal varieties there is some 'classification' via combinatorics, but in the case of non-normal I doubt there is (I never worked with them anyways).

Become a toric fan!.

Answer by Jesus Martinez Garcia for Examples of great mathematical writing

2010-08-10T04:48:15Z

Toric Varieties, about to be published by Cox, Little and Schenck is an unmeasurable amount of joy. It is impossible to get tired of it. Everything is well-bounded and it made me learn as much Algebraic Geometry as Toric Geometry itself. Its introductory sections to Algebraic Geometry before it develops the theory and shows you how to compute examples made me learn more than any dry full theoretic book in Algebraic Geometry. Definetely the best book I have read in two years.

Available at Cox's website

Answer by Jesus Martinez Garcia for Math puzzles for dinner

2010-08-07T06:17:44Z

I think this has not been published yet. Apologies otherwise. I learnt it from Antonio Sánchez Calle in my first year of undergraduate and I had 3 non-mathematicians thinking about it for about 4 hours, so there is a guaranteed success if you tell around :)

5 people are shipwrecked in a deserted island. They find a monkey and lots of coconuts. They spend the whole day collecting coconuts that they keep together and since they are tired they go to sleep. The first person wakes up, attempts to divide the amount of coconuts in five parts, but one of them is spared, so he gives to the monkey. Then he eats one fifth of the coconuts and goes back to sleep.

The second person wakes up and follows the same procedure. He divides the coconuts in five, one is spared and he gives it to the monkey, eats his share and goes back to sleep.

The third, forth and fifth people do the same thing. How many coconuts were there at the beginning? (modulo something, of course).

Comment by Jesus Martinez Garcia

2013-02-26T13:16:40Z

Oh, OK. I thought cusp meant $y^2-x^3$ (never read it anywhere, just from conversations), but I realise that analytically that is also $y^m-x^n$ for $m,n>1$ , which I think is equivalent to what you say. Thanks a lot :)

Comment by Jesus Martinez Garcia

2013-02-22T19:11:48Z

@Jérémy: why must the singularities be cuspidal?

Comment by Jesus Martinez Garcia

2012-12-22T13:13:34Z

I particularly like this interpretation. Did you come with it by yourself or is it well known?

Comment by Jesus Martinez Garcia

2012-12-22T13:11:32Z

Thanks for this answer, but do I understand that you are assuming the field to be finite, or does it work for algebraically closed fields too?

Comment by Jesus Martinez Garcia

2012-12-21T11:40:20Z

Sorry, Piotr, I forgot to mention I was thinking of algebraically closed fields of finite characteristic. I think in that case your example, which is actually quite interesting, does not work. Sorry for being imprecise

Comment by Jesus Martinez Garcia

2012-12-21T11:35:42Z

My bad, I should have stated algebraically closed fields. I did that now. In that case the Fano plane is not applicable.

Comment by Jesus Martinez Garcia

2012-10-16T09:20:31Z

This is great! Thanks a lot :)

Comment by Jesus Martinez Garcia

2012-10-12T17:05:56Z

Update: For the problem I was trying to apply this on, this is not longer necessary (I solved the problem differently). However, it is something I could have applied on other occasions in many other settings (within del Pezzo surfaces, of course). I hope it gets an answer (by someone not necessarily in MO) at some point.

Comment by Jesus Martinez Garcia

2012-10-12T12:36:27Z

Thank you for the clarifications. I think it may be true (i.e. there is a smooth irreducible section in $\vert -K-C\vert$ when the linear system is at least a pencil on a del Pezzo) but I'm still struggling to prove it though. I'll let you know if I make progress. I suspect it is probably not true in higher dimensions, though.

Comment by Jesus Martinez Garcia

2012-10-12T11:16:54Z

@Francesco that's the article I mentioned. @JC Ottem: You are right. I have rephrased this to choose the ample divisor I was thinking all the time on. I think in this case it might be true, but I am not sure how to prove it.

Comment by Jesus Martinez Garcia

2012-10-12T10:41:48Z

In the meanwhile, though, I think if you take in your setting $H=\frac{1}{2}-K$ or even $H=-K$ what I said seems to be true, namely: any general element of $\vert H-C\vert$ is irreducible. I quickly went through Kleiman's article yesterday night, which made me think about proving the following: If $\vert H-C \vert $ is base-point free, then the general member is irreducible. This is not too hard using the base-point free theorem (which probably wasn't available to Zariski). It still doesn't avoid (2) of Extended Bertini. Finally, what does it mean 'composed with the curves of a pencil'?

Comment by Jesus Martinez Garcia

2012-10-12T10:26:37Z

Thank you, Francesco. This is the kind of thing I was thinking on. However your counter-example is not exactly what I was after: $H$ is not a hyperplane section, but rather $F_1+F_2$ is. So in some way, if $B$ is a hyperplane section, you are taking $H=2B-F_1$. Then it is clear you can extract a second $F_1$. My point was to start with a more 'canonical' choice of $H$. The question said 'hyperplane section', but what I actually really want is $H=-K_S$, which for Fano in dimension 2 is either very ample (deg >2), or has Fano index 1 (deg <1). I'll rephrase the question to fit that.

Comment by Jesus Martinez Garcia

2012-10-10T17:34:22Z

That is the joke

Comment by Jesus Martinez Garcia

2012-09-17T16:21:36Z

In WWII there is much more regarding number theory and Turing. Some fiction about it is in the book trilogy Cryptonomicum

Comment by Jesus Martinez Garcia

2012-04-10T15:10:43Z

It was when I posted this :)