From: Michael Stillman <mike.stillman@mac.com>
To: Dan Grayson <dan@math.uiuc.edu>
Subject: schubert2
Date: Thu, 05 Mar 2009 01:49:01 -0500

I talked to Sheldon yesterday, showing him Schubert2.  He hadn't seen  
it yet.  He asked about the following example.  I'm going to try to  
figure out if we can do this one now, or not.  (from demo.m2).  I  
think we can, but I need to figure out the syntax...

----  
#-------------------------------------------------------------------------
---- # Riemann-Roch formulas.
---- # Line bundle O(D) on a threefold.
---- >
---- > variety(X,dim=3,tan=sheaf(3,[-K,c2,c3])): # traditionally, -K is
---- >                                           # used instead of c1
---- > chi(o(D));
----                         3          2          2
----        integral(X, 1/6 D  - 1/4 K D  + (1/12 K  + 1/12 c2) D -  
1/24 K c2)
----


=============================================================================

From: Michael Stillman <mike.stillman@mac.com>
To: Dan Grayson <dan@math.uiuc.edu>
Subject: schubert2
Date: Thu, 05 Mar 2009 02:02:36 -0500

The example in the last email can almost be done as:

X = base(3, Bundle=>(L,1,D), Bundle=>(T,3,b))
X.TangentBundle = T
chern L
part_3(chi L)


Problems:
(1) He would like to name the first chern class of L to be D, not D_1.
(2) He would like chern T = 1 - K + c_2 + c_3
(3) chi L gives elements of degree higher than 3 !  This is a bug, as  
X has dimension 3.

(1) and (2) are perhaps cosmetic, but perhaps we have the ability to  
name each chern class now?  I couldn't easily find it.

Perhaps we need a way to introduce generators into the intersection  
ring of X that have higher degree than 0.  That way we could first  
define X with these values (and perhaps with whatever relations they  
satisfy), then we could create the two sheaves afterwords?

Or, once again, maybe we can do this?!

I want to sit down and document what we have so far...  At least the  
part that won't change.

David E and Joe Harris are working on a book on intersection theory,  
and they would like to do Schubert2 examples throughout.  I think that  
this would be great, and that we should improve Schubert2 as much as  
we can for that.  They plan on working hard on their book this fall  
(although we can ask them next week what they would need/want from us  
at various times).


=============================================================================
From: Michael Stillman <mike.stillman@mac.com>
To: Dan Grayson <dan@math.uiuc.edu>
Subject: schubert2
Date: Thu, 05 Mar 2009 02:11:18 -0500

Here is another way to get that formula, although this is perhaps a  
bit tedious:

A0 = QQ[D,K,c_2,c_3, Degrees => {1,1,2,3}]
A = A0/truncate(4,ideal vars A0)
X = abstractVariety(3,A)
L = OO(D)
T = abstractSheaf(X,Rank=>3,ChernClass=>1-K+c_2+c_3)
X.TangentBundle = T
part_3(todd X * ch L)


-- note: chi L fails, as this doesn't know how to compute integral in  
this case...
-- this is why I needed to do the last line.

