2015 AG Institute in Utah: Day 7

(As has consistently been the case this post is one day late, and today is actually Day 8. For background on the conference and my time in Utah see: Day 1 and Day 3.)

Today marks one week into the three week mega conference in algebraic geometry! Things have begun to settle in a bit, and this weekend was a nice break from the usual five+ hours of talks a day. That said the weird thing about this conference is that since it is 3 weeks long people are always coming and going. It’s almost as if the conference is actually three miniature conferences in one.

2015-07-16 17.57.41

This weekend has also given me a bit more time to explore Salt Lake City. Here are a couple things I have noticed:

  • Everything in this city feels fair away, and not just because I am living at the furtherest edge of the city on the foothills. (It is actually about a 3 mile walk from my place to the edge of what appears to be the start of the city.) I mean that the blocks seem really long and streets very wide, and so getting from place to place takes a while. Someone told me that I am not crazy and Salt Lake does in fact have abnormally large block and wide streets, and apparently this dates back to the city’s founding and Brigham Young. Apparently Brigham Young wanted each block to be exactly 10 acres with streets being something like 100 feet wide. He thought each family should essentially have a small farm or something. While this might have been great back in the 19th century when Salt Lake was being founded it seems less ideal in the 21st century. As a plus Brigham Young did have a pretty sensical way to name streets as everything is essentially labeled as if it were on the cartesian plane.
  • In addition, to the large blocks Salt Lake also seems really spread out — did I mention I am like 3 miles to the start of down town. Luckily there is a pretty nice public transportation system featuring buses (haven’t used them) and a train (have used it). The train is incredibly nice. It is quite, smooth, and pretty clean. Moreover it covers a large amount of the city. (This is how I’ve been getting into the city.) That said it does have a few huge down sides: 1) It is like 25-3oish minutes between trains and 2) The train stops at 11pm on Friday nights!! Really 11pm on a Friday! Does Salt Lake have any nightlife? All of these things combined have meant that for the most part going out at night is a huge chore and has generally not seemed worth the effort.
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Strange things you see from the train.

  • The landscape in Salt Lake is amazing with mountains on all sides providing great views. However, the city itself — or at least the bits I’ve been to — are not that pretty. Maybe it is due to the fact that all of the roads seem to be 8 lanes wide and every business has a huge parking lot, but walking downtown is just not enjoyable.
Salt Lake City From Campus

Salt Lake City From Campus

Finally here again are some of the interesting things I learnt / heard during talks on Day 4 and 5. (Once again a quick disclaimer that there are most likely errors in what follows and those are due entirely to myself.)

  • Non-Negativity and Sums of Squares (Greg Smith): Given a real polynomial f\in \mathbb{R}[x_0,\cdots,x_n] we say that:
    • f is non-negative iff f(x)\geq0 for all $latex x\in S:=\mathbb{R}^{n+1}.
    • f is a sum of squares iff f=g_1^2+\cdots+g_s^2 for some $;atex g_i\in S$.

    Having made these definitions there are two fairly obvious, but important, observations one can make: 1) the definition of non-negative only makes sense for real polynomials and 2) a sum of squares is clearly non-negative. In light of observation 2) one might ask when the converse holds i.e. when is a non-negative polynomial necessarily a sum of squares. It turns out one of math’s most influential question asker — David Hilbert — asked essentially this question as his 17th question back in 1900. As is the case with many of Hilbert’s problems this one was solved in the form of the following theorem due to E. Artin:

    Theorem (E. Artin): A polynomial f\in S is non-negative if and only if there exists h_1,\ldots, h_s, g_1,\ldots,g_r\in S such that:

    f(h_1^2+\cdots+h_s^r)=g_1^2+\cdots+g_r^2 \quad \quad (1).

    While this theorem nicely answers the question at hand, it does so in an entirely non-constructive way — the proof apparently uses the axiom of choice — and so one could wonder if there is an effective version. That is if f is a non-negative polynomial then is there an upper bound on how large the degree of the h_i we might need to check for the above theorem. It turns out a 2014 paper of Lombardi, Perrucci, and Roy provides such a bound:

    Theorem (L,P,R): If f\in S is non-negative polynomial such that \deg(f)=2j then there exists h_1,\ldots, h_s, g_1,\ldots,g_r\in S satisfying (1) such that:

    \deg(h_i)\leq 2^{2^{2^{j^{4^{n+1}}}}}.

    Ok so that bound while theoretical effective is in practice no where near useful. Moreover it would seem mildly ridiculous if that bound was actually sharp. In his talk Greg Smith discussed how this result can be improved.

    The general idea behind Smith and collaborators’ — whose names I cannot find in my notes — work is to utilize the power of algebraic geometry. Most of the work done on this question to this part goes via the use of semi-algebraic sets and not varieties. However, in many ways varieties are much better understood, and more intrinsic, than semi-algebraic sets. Thus, the idea is try and rephrase this question in terms of varieties and then leverage the machinery that is algebraic geometry.

    In order to do this Smith et al. first have to rephrase the problem in terms of algebraic geometry. From now on X\subset \mathbb{P}^n_{\mathbb{C}} is will be a real projective sub-variety. By real sub-variety we mean that X is defined by real polynomials. Moreover we require that X(\mathbb{R}) be Zariski dense in X. While this last condition might seem restrictive it is actually equivalent to X having a smooth real point. Finally let R be the coordinate ring associated to X. With this set up we say f\in R_{2j} is:

    • non-negative iff its evaluation at each point in X(\mathbb{R}) is greater than zero.
    • a sum of squares iff there exist g_1,\ldots, g_r\in R_{j} such that f=g_1^2+\cdots g_2^2.

    As a quick reality check you should think why the definition of non-negativity makes sense. Remember f is not necessarily a function and we are working with projective points. In this set up they are able to obtain the following theorem:

    Theorem (Smith, et al.): Let X\subset \mathbb{P}^n be a curve of degree d and arithmetic genus p_a. If $f\in R_2j$ is non-negative then there exists a sum of squares $h\in R_2k$ such that $fh$ is a sum of squares such that:

    k\leq \max\{d-n+1, \text{ceil}(2p_ad^{-1})\}.

    Notice that this bound is way way more reasonable looking that the bound found by Lombardi, Perrucci, and Roy. Moreover Smith, et al. have been able to show that their bound is sharp!! Also notice that the bound is entirely independent of f and depends solely on the geometry of X!!! Sadly they are not able to extend this result to higher dimensions. However, they do know any optimal bound in higher dimensions cannot be independent of f.

    The idea behind the proof seemed a bit complicated, and I don’t want to go into the sketch presented in the talk. However, I will say this it utilizes everyone’s second favorite type of cone. (For those who didn’t know what that I was assured it was spectrahedral cones.

  • Geometry and Syzygies of Curves (Robert Lazarsfeld): Robert Lazarsfeld also gave an amazing talk highlighting the connected between the geometry of a curve and its syzygies. The general set up is very similar to that of Gavril Farkas’s talk (see Day 3). Let C be a smooth curve of genus g\geq2 and L a line bundle on C such that \deg(L)=d>>0. Now |L| defines an embedding of C into \mathbb{P}^{d-g}, and we will let S=\text{Sym} H^0(L) be the coordinate ring associated to C. Given another line bundle B on C we define R(B, L) to the S-module:

    R(B, L)=\bigoplus_{m\geq0} H^0(C, B+mL).

    If we let $B=K_c$ be the canonical canonical class it turns out R(K_C, L) is finitely generated as an S-module by H^0(C, K_C). In fact, the multiplication map \mu: H^0(C, K_c)\otimes H^0(C, L)\rightarrow H^0(C, K_C+L) is surjective. Taking the kernel of this map we get the following complex:

    S(-1)^N\rightarrow S^g\rightarrow R(K_C, L) \rightarrow 0.

    Thus, the question becomes: When is the above complex exact? Put differently when is the above a linear presentation for R(K_C, L)? It turns out that Mark Green gave an answer to this question:

    Theorem (Green): If C is a smooth projective curve of genus g and L is a line bundle on C such that \deg(L)>>0 then:

    S(-1)^N\rightarrow S^g\rightarrow R(K_C, L) \rightarrow 0.

    is exact if and only if C is not hyper-elliptic.

    The question Lazarsfeld went on to discuss when does this pattern continue? That is if E_{-}\rightarrow R(B, L)\rightarrow 0 is a free resolution of R(B, L) when can we guarantee that E_{p}=S(-p)^{n_p} for some range of $p$? With this question in mind we say that B satisfies M_l iff E_p=S(-p)^{n_p} for all 0\leq p \leq l.

    With all of this set up the main conjecture that Lazarsfeld discussed was the following:

    Conjecture: If C is a smooth projective curve of genus g and L is a line bundle on C such that \deg(L)>>0 then K_C fails M_l if and only if \text{gon}(C)\leq l+1.

    Recall that a curve C has gonality at least l if there is a degree l+1 map from C to \mathbb{P}^1. Thus, this conjecture explicitly connects the geometry of a curve to the its syzygies. In this light it has a very similar flavor to the Green-Lazarsfeld Secant Conjecture discussed in my post on Day 3.

    This conjecture turns out to be true!! In fact a stronger conjecture relating the vanishing of syzygies to the ampleness of the line bundle of interest is true!! However, this post is already getting long so I will spare righting about that. As Lazarsfeld explains it the proofs of these conjectures were hanging in plain sight, however, they only were noticed following Voisin’s work on Green’s Conjecture.

    Overall this talk was awesome, and I am really hoping to learn more about this area at some point in the near future. In fact I found it so exciting I purchased this book on Koszul cohomology in algebraic geometry. As a final note one subtle thing regarding all of these theorems is that the degree of the embedding line bundle needs to sufficiently positive. However, from the talk it wasn’t clear what sufficiently positive means. Is there an explicit bound above, which these theorems work, or is it just an eventually type statement?

Well that’s all for now. Stay tune for another update either Wednesday or Thursday!!