Lecture 1:  Toric Ideals

1) Consider the system of equations

    a*x^2*y + b*x*y^2 + c = 0
    d*x^2*y + e*x*y^2 + f = 0 
     
   Show that for generic (a,b,c,d,e,f), this has 3 solutions.
   Give a set of inequations describing when this has 3 solutions.

2) Let \calA be a set of exponent vectors, and \varphi_\calA the
    corresponding map from the n-torus to k^\calA.  Show that \varphi_\calA is
    injective if and only if the integral linear span of \calA is the full
    integral lattice.

3) Pick a sparse system of equations, and compute the degree of its
   ideal.   Using elimination theory, compute the degree of the
   corresponding toric variety (for this, first homogenize the problem).

 Example of the second part:  

     a*u^3 + b*s*t*u + c*s^2*t + d*s*t^2 = 0

   Consider the map from the 3-torus (C^*)^3 -> C^4:
     (s, t, u) |---> (u^3, s*t*u, s^2*t, s*t^2)
   Using elimination theory, we may compute its ideal, J:

   ring R = 101, (s, t, u, w, x, y, z), dp;
   ideal I =  w - u^3,  x - s*t*u,  y - s^2*t,  z - s*t^2;
   I = eliminate(I, stu);
   ring S = 101, (w, x, y, z), dp;
   ideal J = std(imap(R,I));
   J;
   dim(J);
   mult(J);

4) Show that I_A+ is the largest homogeneous ideal contained in I_A

5) Recall the `bad' triangle of the example (after Koelman's Theorem) 
   in the lecture.  Find more examples of polygons whose toric ideal are  
   not generated by quadrics.

6) Consider the map \varphi_\calA from a n-dimensional torus to P^\calA.
   Show that this is injective if and only if \calA affinely spans Z^n.

7) Compute the Erhart polynomial of a few polytopes.

8) Use Kouchnirenko's theorem to compute the volume of some non-trivial
   polytope  (e.g a cylic polytope = conv { g(a1), g(a2),..., g(an) }
   where g is the rational normal curve in d-space with ai in Z. Try d=2
   or 3)
 
9) Consider the flat toric deformation of the twisted cubic [1, t^2-1, t^3-t, t]
   given by (s, [1,x,y,z]) |---> [1,x,y,sz].   What is the limit scheme
   as s approaches zero?

10) Let A = | 1 1 1 1 | be exponent vectors giving the rational normal curve.
            | 0 1 2 3 |
    There are 9 types of liftings for this polytope (a line segment of length 3).
    For each give
    a) A weight vector w inducing it.
    b) The triangulation and facet sets F_A
    c) The initial ideal lt_w(I_A).
    Explain the scheme of each initial ideal in c)

   Are these all the possible initial ideals.

Exercises on Mixed Volumes (d'apres I. Soprounov)

1) Mixed Volume and Bernstein's Theorem

 a) Show that MV(P, I) = V_n-1(pr(P))*V_1(I), P = n-polytope, I = 1-polytope,
    and pr = projection along I

 b) What is MV(P_1, P_2, ..., P_{n-1}, pt) ? pt:=point

 c) Find MV(I_1, I_2, ..., I_n), where I_i is a line segment in R^n

 d) Prove that V(P_1,\dotsc,P_n) = 0 if and only if there is some
    subset of the polytopes size k with dimension of the sum of these polytopes
    is less than k 

2) Prove that the number of solutions of a binomial system:
      A_i t^{a_i} + B_i t^{b_i} = 0 i=1,..,n
   equals | det(b_i-a_i) |.


Enumerative Geometry

1) Verify the assertion that through any three mutually skew lines in P^3
   there is a unique ruled quadric surface.  Describe both rulings.
 -> Hint: You can assume that the three lines have a particular form:
     [*:*:0:0], [0:0:*:*], and [a:b:a:b].

2) Do #1 via a symbolic computation.  Pick three random lines in 3-space
    (represented as triples of colinear points {a_i, b_i, (a_i+b_i)/2} )
    and compute the equation of the unique quadratic surface they lie on.
    (solve the equation Q(points on the lines)=0, where Q is an arbitrary
    quadric.)
   Next, use elimination to compute the family of lines lying on the quadric Q.
   As a bonus, describe this in Pluecker space as the intersection of the 
    Pluecker quadric with a P^2.
    
3) Let F : P^1 --> P^{r-1} be a map defined by forms f_1,...,f_r of degree n-1.
    Choose a local parameter s at a point x in P^1 to define the derivatives
    F', F'', ... of F.  Show that the linear span
    < F(x), F'(x), F''(x), ..., F^(i)(x) > is well-defined and independent of
    choice of local parameter.

4) Recall the system of local coordinates U_\alpha for G(r,n) described 
    in the lecture (given by matrices with I_r places in columns \alpha 
    and Mat_{r \times (n-r)} in the other columns).  Prove that this is 
    indeed a system of local coordinates for the Grassmannian.
    (The coordinate changes are regular functions and the charts cover
    the Grassmannian.)