About six months ago I asked people for opinions about digital pens, and did my own research, so I decided to pick up a LiveScribe Echo based on what I’d heard, and the campus computer store has a 10 day return policy (so long as packaging is intact) so I decided to give it a try.  For those who are impatient, the short version of my review is: the digital pen is good, and I’m keeping mine, but there are a few improvements that would go a long way to mainstreaming, so I give it a 4/5.

I’ve used the pen for almost six months of my regular activity, including doodles, mathematical scratchwork, seminar notes, and writing based hobbies.  Here are the pros and cons:

Pros

• The detection is actually much better than I had hoped.  The pen captures my writing very well, from fairly small handwriting to large strokes.
• On the purchasable notebooks, the dot paper is only barely noticeable and not distracting at all.  The free ones that you can just print out aren’t as good, but you can print out 100 pages of the paper at a time that won’t cost you a thing (assuming access to an early modern laser printer)
• The software is very easy to use.  I just plug my pen in, open the LiveScribe desktop, and it gives me a list of all the notebooks I have active with the pen, and loads all the pages I’ve written (each page of each kind of notebook has a unique dot pattern, which is aperiodic, so a small sample tells the pen in what notebook you’re working, on which page, and where on the page) and then moving them to custom notebooks, like one for each date or subject, is just click and drag.
• PDF output
• The audio recording is pretty good, and it’s nice that it will synch the audio to your penstrokes, letting you see how you took notes during a lecture.
• The search feature is fairly good.  If you type in a word, it will use a sort of shotgun approach to finding instances of your having written the word, which is about as good as you can expect without going into high end handwriting recognition (and there is apparently an app you can purchase that improves the recognition a LOT, but I haven’t bought it)

Cons

• The pen is a bit too large.  I’ve heard people complain that it’s like writing with a magic marker, which is a bit of an exaggeration, and there’s no problem for short bursts, but over the length of a seminar talk, I found that my hand was slightly sore from holding it differently from a regular pen.  Now, I admit that my hand is sometimes slightly sore from a regular pen if I’m writing a lot, which I certainly was, because I had started livetexing some time ago.  However, I think that this problem will be solved by using the pen more, and long term I expect the pens to slim.
• You need to use their special dot paper.  This isn’t a MAJOR problem, except that there isn’t that much in the way of selection.  I personally do most of my work on unbound, unlined paper, or bound quad-ruled paper.  Most of the paper and notebooks they have are college ruled, which, and this may just be a personal idiosyncrasy, but I hate doing math on lined paper.  The free paper is all lined, and there are unlined journals that can be purchased, but there’s a definite lack of variety.  I would like to be able to print out my own unlined paper, or quad-ruled, which there seems to be no way to do.
• The pen rolls.  It rolls a lot.  It’s a bit too symmetric.  They should add a clip or something to the top, just something that will allow it to rest, rather than roll, on slightly slanted surfaces

Overall, I like the pen a lot, and it’s caused me to livetex a lot less, partly because it has a much better battery life than my laptop does and it’s a bit quieter to use than a keyboard.  There’s definitely room for improvement, but the things that annoyed me I’m noticing less and less often the longer I use the pen.

We, in fact, can describe this abelian variety in many ways. The simplest is probably that we have a pullback map , and we actually get a short exact sequence of abelian varieties , and is called the Prym variety associated to the cover.

Another description is that we can define a map , given by . Then, we look at the fiber over zero. The kernel here actually breaks up into two components. The component containing is isomorphic to , and the other is just a translation of .

So the first real theorem about Prym varieties is the following:

Wirtinger’s Theorem: The theta divisor on induces twice a principal polarization on .  That is, .

So if we set the space of unramified double covers of curves of genus , and the space of principally polarized abelian varieties of dimension , then we have a map .

This leads to one of my favorite theorems

Theorem: The closure of the image of contains the Jacobians of genus .

To see this, we’ll use what are called Wirtinger covers.  If , then is a stable curve of genus .  We can then set .  Then we’ll need to work out the Jacobians of and .  For , a line bundle is the same as one on , along with a nonzero complex number, that is, a map identifying the fibers over and .  So .  Similarly, we have , and we get vertical maps given by multiplication, norm and tensor product.  This then presents where is a finite group, and so the connected component of the identity is .  So allowing the double cover to degenerate gives Jacobians!

Thus, the Prym map is closely tied to the problem of figuring out which abelian varieties are Jacobians (for more details, see my ICTP talk).

Let and . Then, take complementary of dimensions and , that is, , and their span is all of . Now, choose and rational normal curves. Finally, choose an isomorphism .

A (2d) rational normal scroll is . That is, it’s the union of the lines from one rational normal curve to another. This depends only on the numbers , and not on the choice of subspaces, rational normal curves, or isomorphism.

Now, if , then the lines used in the definition are the only lines on , and we’ll call those the lines of the ruling. However, for , we can get some classical examples that we’ve seen before. is given by taking two skew lines in , and takes the unions of lines between them, so this gives a quadric surface. The next simples, , turns out to be what you get when you embed the plane into as the Veronese surface, and then project to from a point on the surface.

We can generalize the construction even further: set such that , and pick complementary subspaces. Next, pick rational normal curves, and isomorphisms. Then the -dimensional rational normal scroll is . This is also called the rational normal -fold scroll.

Two quick examples of these are that is the Segre embedding of into , and more generally, times is the Segre of into .

So, now that we’ve talked about rational normal scrolls, the following theorem can be proved:

Theoem: Let be an irreducible and nondegenerate variety of dimension .  Then the minimum possible degree of is and the possible varieties with this degree are:

1. Quadric hypersurfaces
2. The cone over the quadratic Veronese
3. Rational normal scrolls

This isn’t trivial to prove, but is VERY useful.  Here are a few consequences:

• If is a rational normal -fold scroll, then a hyperplane section is a -fold scroll.
• Projection from a point of a scroll is a scroll.
• The examples above actually can be proved from this theorem
• Rational normal curves are minimal curves

There are quite a few other consequences of this, and this is connected to classical Castelnuovo Theory, and to some much more recent work of Pareschi and Popa, generalizing this to abelian varieties.

We can actually compute what it is for . An matrix has rank 1 if and only if we can write it as where and are vectors. Working out the equations then gives us , which should be familiar, this is the image of the Segre map on .

The story of determinantal varieties seems to be the story of replacing the entries of a matrix of variables with polynomials. The next thing to try is to let be a matrix of linear forms on which don’t all vanish simultaneously. Now, we set to be the set where has rank at most . We call this a linear determinantal variety. One way to interpret this more geometrically, and using the earlier definitions, is that gives a map , and then the linear determinantal varieties are the pullbacks along of the . This suggests, to me, that the name “generic determinantal variety” isn’t quite right (though it’s what Harris uses in his “Algebraic Geometry: A First Course”), I think that perhaps universal is a better modifier, as at least when all the entries are of the same degree, we can get the varieties in this way.

Rational normal curves are of the above type, though. Given then is the rational normal curve of degree .

However, not all determinantal varieties are made from matrices where the degrees are constant. In fact, if is a matrix of forms on , then we get a determinantal variety so long as can be written as for some vectors and . Then the minors will all be homogeneous, and so their zero locus is a determinantal subvariety of .

So, now let’s look at one type of variety that ISN’T a linear determinantal variety, so that this concept it actually useful: the generic surface of degree . These are given by the determinants of matrices of linear forms that don’t identically vanish. Call the space of such matrices . Then we can left or right multiply by scalar matrices and it doesn’t change the determinant, except a scalar. So we can quotient and get , which has dimension .

However, the dimension of the space of degree surfaces is .  So then for already, the generic surface is not determinantal! Perhaps more interestingly, cubic surfaces ARE, generically.  There’s a lot more to say, but honestly, I only have bits and pieces.

I’ve looked at a couple of them, and I’m really not sure what their specific pros and cons may be, and I have no real way to try them out directly at the moment.  There’s the LiveScribe pens, which require their magic dot paper stuff, which is a drawback that I’m not sure how annoying it would be, plus I don’t know how useful sound recording would be, though I can see myself maybe using it at some point.  The other brand I’ve looked at is SolidTek’s DigiMemo, which is a bit bulkier being a clipboard, and amazon reviews suggest it’s finicky.  Is there another brand I should look at? I’m mostly looking for a way to nicely digitally archive all of my scratchwork (my collection of notebooks is expanding too fast!) and also maybe taking notes at seminars, conferences and the occasional advanced course.

So, anyone? I figure that any math person who has a digital pen or has at least considered them will have a lot of the same uses in mind as me, and so I’m very interested in opinions.

Let’s start with some definitions.  The set is the set of where is a genus curve and are distinct.  Now, sadly, is not a variety.  It can be realized as a smooth stack, however, but that’s a bit more than we are going to even try to handle.  Instead, we’ll quote a very good, powerful and classical theorem, which is true (and clearly so, because I am using a boldface for the word theorem):

Theorem: There exists a unique normal quasiprojective variety such that it is in bijection with and if is a proper smooth map whose fibers are genus curves and are disjoint sections, then the induced map of sets is a morphism of varieties.

Rather, the above is true when .  When it’s smaller, then the space has expected dimension negative (this only happens for genus 0 with at most 2 points, and genus 1 with no points) and we really need the stacky interpretation to make sense of things.

So why is this the dimension? The computation is fairly quick: first off, each point is a degree of freedom, so , and is just deformations of a curve.  It can be shown that the tangent space to at a curve is just , so we must compute this dimension, which by Riemann-Roch is for .  For , we need to check things carefully because of the aforementioned stacky problems (it amounts to positive dimensional families of automorphisms for these curves), but the result works out.

We’re really only going to have one example, and then the others will follow from it.  We’ll directly compute for .  There’s only one genus zero curve, and that’s , so we’re looking at sets of ordered points in .  The automorphisms let us take the first three to , so we have points to pick, and there are no automorphisms left once we’ve done this, so , because the points are distinct.

Next, we’ll look at , as this is the next simplest that has positive dimension.  There are a few ways we can attack this.  The simplest is by using Weierstrass form and the -invariant, and this works for characteristic not , and we’re going to imagine ourselves over , so it’s fine.  Every elliptic curve can be written as for some , with .  However, this representation isn’t unique.  So whatever our curve is, it’s covered by .  There’s a function , and it takes the same values at , , , , , and , which all give the same elliptic curve! Even better, no other will give the same curve, so is a surjective map.  And because can take any value, it shows that .

A slightly different presentation is that every elliptic curve can be written as a double cover of ramified at four points.  These points are unordered, and so we can look at quartic polynomials on with no repeated roots .  However, automorphisms of let us take the roots to , so we have and its discriminant is nonzero, and we have to identify the same ‘s as above.

This approach is fruitful for identifying a certain subvariety of for .  This is the locus of hyperelliptic curves.  These are the curve which double cover and thus are uniquely determined by their branch points.  For a genus hyperelliptic curve, we have ramification points.  The second approach above is predicated on the fact that for , every curve is hyperelliptic.  We’ll denote the hyperelliptic locus by .

Now, it’s a fairly easy theorem that , so we should be able to handle finding a description of this family of curves.  By the above, they branch at six points, so we have to look at six distinct points in , which is an open subset of the sextic polynomials.  Then, we need to take .  We can achieve this by looking at ordered sextuples of points on , and having an extension of by (or perhaps extension in the other direction? I never quite got that bit of language) act by first permuting the points, and then transforming so that the first three are .  This can actually be calculated directly, and it’s a quotient of .  The points that need to be identified are the ones where, if is a primitive fifth root of unity, we have gives the same curve as .  It’s very straightforward to compute the invariants, and they are , and give a map which factors through the quotient, thus giving us inside of .

In general, we have that can be written as a quotient of , though computing this quotient is not always easy.  This is also less useful in higher genus, because starting in genus 3, there are nonhyperelliptic curves.  In fact, the moduli space can be described in two pieces: the hyperelliptic locus can be worked out as above.  For the nonhyperelliptic curves, the canonical map gives them as quartic plane curves, so we have the open subset of the linear system of quartics consisting of smooth curves, and we have to take , the automorphism group of the plane.  This is plainly unirational (for those who know the term), and I vaguely recall seeing a paper proving that it’s actually rational, but I can’t find it at the moment.  So starting at , these moduli spaces can be very hard to construct directly, and that’s a big part of why this subject is interesting, and why it is so difficult.

A simplicial complex is a nice type of topological space.  Take a collection of points, called the vertices , and define to be sets of elements of which are -simplices, that is, every subset of elements of an element of is an element of .  We can construct these geometrically by looking in a really large Euclidean space, picking points, and for each simplex, taking their simplicial span, that is, all linear combinations of the points with coefficients nonnegative and summing to one.

Now, we start with the algebra.  Given a simplicial complex , we have a ring , which is the free -algebra on the vertices.  We can take each element of to be a monomial of degree in the vertices, in fact, every possible -simplex is such a monomial, and squarefree!  So we define an ideal to be generated by the non-faces of , and we define the Stanley-Reisner ring of to be .

So now, a couple of examples.  The first is the trivial example: the -simplex.  For the -simplex, every subset is a face, and so , so .  Now, a less trivial example is in order: the octohedron.  It has six vertices, and we can label them so that the non-edges are , and these will be the generators of .

So now, we define the -vector of , which has for the number of -faces, and we consider to be a face.  So the octohedron has .

Next, we look at polytopes.  These are the convex hulls of a finite collection of points in some Euclidean space.  We’ll actually care only about the case where the boundary is a simplicial complex, we’ll call these simplicial polytopes.  We even get a couple of nice formulas: there’s Euler’s formula that for a simplicial polyhedron in , we have and , and so determines the -vector.

Now, it’s clear what date the -vector describes, but we’re going to transform it into a much less obvious form, and from that, pull a nice theorem out of thin air.  The -vector is defined to be and can be recovered as .

So…where does this come from? This is just a strange thing to do.  Lets look at itself.  We want to try to make a graded free resolution (which we’ll treat largely as a black box).  This amounts to there being an exact sequence with maps of degree 0 which is free modules, except the last term.  For the octohedron, we have .

For the octohedron, the -vector is , which you can see are the exponents here, and that’s not a coincidence! But more fundamentally, we define the Hilbert series of a graded module to be , and this is additive in exact sequences.  So the resolution above lets us compute , so the -vector is the set of coefficients of the reduced Hilbert series!

Now, it’s a nontrivial theorem that if is a -polytope.  So then, we take to be such a polytope.  Then it defines a simplicial projective toric variety , and it can be proved that .  So this gives a weaker version of Poincare duality for “nice” but possibly singular toric varieties!

Fix a curve .  A vanishing thetanull on the curve is a theta characteristic with .  This occurs if an even theta characteristic is effective, or if an odd theta characteristic gives a map to the plane.  This is actually a rare occurrence, and a generic curve doesn’t have a vanishing thetanull.  So now we’re restricting to generic.

A theta characteristic on such a curve is effective if and only if it is odd.  On top of that, as , the divisor associated to can be doubled to give a canonical divisor.  What does this mean geometrically? It means that if we can the canonical embedding of the curve, then is a hyperplane section.

Now for the nice classical consequence: let .  Then the canonical curve lies in , and is degree 4.  So we have a smooth plane quartic.  Generically, there are no vanishing thetanulls, so every divisor such that is the intersection of with a line comes from a unique odd theta characteristic, and is uniquely determined by it.  So is a degree 2 divisor, so the line intersects the curve in two points.  As , that means that the line is tangent to the curve at both points.  Thus, for a generic curve of genus (that is, a generic quartic plane curve), the number of bitangents is exactly the number of odd theta characteristics.  Last time, we computed this number: .

This 28 has interesting connections to many things, but I’ll only mention one.  They are the zero locus of a quadric of Art invariant 1 on .  If we projectivize, so that we have a quadric on , then the locus consists of 27 points, and if we say that two points are incident if they lie on an isotropic line, then we get the incidence relation of lines on a smooth cubic surface! It’s even better than that.  If we take a smooth cubic surface in , and project to the plane from a point not on any line, then we get a double cover of the plane branched at a smooth quartic.  Then, the 27 lines go to 27 of the bitangents, and the other one is the exceptional locus of the blowup! So, odd theta characteristics correspond to bitangents, and for quartic curves, these correspond to the lines on a cubic surface, plus a point not on any of the lines.

We start by denoting the set of theta characteristics on a curve by .  The first nice property of this set is that it’s a torsor over the points of order two on .  We have a natural action of on by .  This will still be a theta characteristic, because after squaring, .   This action is free, and it is transitive, because if are theta characteristics, then squares to the trivial line bundle, and so they differ by some point of order two.

This argument tells us that there are theta characteristics of .  We then split them into two types, even and odd, distinguished by the parity of with a theta characteristic.  But how many of each are there? We’re going to spend the rest of this post computing these numbers.

A quadratic form on a vector space is a function such that for all , we have , and such that the map given by is bilinear.  Then, .  As we’re going to be looking in characteristic two, this implies that , and so is a symplectic form.  Now, we fix a basis of , and define .  Then we can write .  We restrict to looking only at nondegenerate , where .

Now, as is symplectic and is nondegenerate, we must have , and we can choose a basis such that , so we can write .  We only care about the field of two elements, in fact, so every element is its own square root, so .

We can define a chosen quadratic form by .  Then, given a form and a vector, we can write , so the quadrics are a torsor over the vector space itself.  We define the Arf invariant by .  It turns out that, as any can be written uniquely as , the Arf invariant is , and this value is called the parity of the form, that is, even or odd.

So now, we can compute the number of even and the number of odd quadratic forms.  It’s just and .   Computing the number is actually a straightforward induction argument.  If , then the cardinalities are and .  In general, they are and .

So, how does this connect to theta characteristics? Fix a theta characteristic .  Then it defines a quadratic form on the -vector space , defined by .  These will be even or odd based on whether is, and so we’ve computed the number of even and odd theta characteristics!

We start with a problem: is there a natural map ? The answer is no, sadly.  However, all hope is not lost! Nay, there IS a natural map , given by taking each point to the divisor represented by .  For , this map is an isomorphism onto its image (injectivity is easy, as two points being linearly equivalent implies genus zero, smoothness of the image is a bit less obvious, but not hard).  So we know that sits inside of , but that requires some identification of with .

In fact, for all , we have a map given by taking .  This even factors through , because the addition on is abelian.  We will call these the level Abel-Jacobi maps, and if , we call it the Abel-Jacobi map, and denote them by , and drop the subscript for .  We call a curve in an Abel-Jacobi curve if it is of the form where .

Moving to higher dimensions, we denote the image of by , which is the locus of effective divisors in .  For any , the fiber is the set of all divisors linearly equivalent to it, and thus is a projective space, naturally identified with .

We can define a function , which is just the dimension of the fiber over .  This function is upper semicontinuous, that is, the loci where are all closed.  We define these loci to be , that is, is the locus of all complete linear systems of degree and rank on .  We’ll return to these later.  For now, we’ll study .

Here, the codimension is 1, so we have a divisor, as is a smooth variety.  The Riemann-Roch formula tells us that for , we have , so is an involution of , which we’ll denote by .  We’ll use the same letter for the involution applied to arbitrary points of .  Now, if we take a symmetric theta divisor, that is, as before, such that if then , for , then we get a nice theorem:

Riemann’s Theorem: There exists a constant such that .

This theorem gives us a geometric model of the theta divisor, which we can now construct explicitly from the curve.  It’s the first step in a program that ties together the geometry of the pair with the geometry of , and culminates in the Riemann Singularity Theorem and the Torelli Theorem.  We’ll pick up these threads later.

For now, we’re going to look more closely at .  If we denote translation by as , then we can write Riemann’s Theorem as .  Now, looking at , we note that symmetry of , Riemann’s Theorem, and the fact that is -invariant, tells us that , and so .  Thus, is a line bundle satisfying .  Such line bundles are called theta characteristics, and these turn out to be extremely closely tied to the geometry of the curve .