This post contains qualifying exam questions on algebraic geometry.
- Let be a smooth complex projective curve of genus . For any , prove the following statements:
For any , there is always a nonconstant rational function on which is regular everywhere except for a pole of order at . (By Riemann-Roch we have for ).
(Weierstrass gaps) There are exactly numbers such that for each , there is no nonconstant rational function on which is regular everywhere except for a pole of order at . (The numbers are between and and by Riemann-Roch each increment is at most 1.)
Let be a Zariski closed subset. Define the Hilbert function and the Hilbert polynomial . (Let be the defining homogeneous ideal of ; then can be defined as the codimension of the -th graded piece. For large values of this is a polynomial in , called the Hilbert polynomial of .)
Suppose . Show that . (It suffices that for any , we find a homogeneous polynomial of degree that vanishes at all except . We simply take product of linear forms that vanishes at but non-zero at all other points.)
Again, suppose . Show that unless is contained in a line. (Again as in (b) but this time we need to find a homogeneous polynomial of degree that vanishes at all but one point. The idea is to if not all points are colinear, then for any , we can find and a linear form that vanishes at and but not . Then we simply take the product of with other linear forms that vanishes at for .)
Let be a variety of dimension , let be the Grassmannian parametrizing lines in and let be the locus of lines contained in . Show that with equality holding only if is a -plane in . (Consider the map sending the line connecting and and comparing dimension.)
Show that Grassmannian is a proper scheme. (use the representability criterion, see here for answer; for properness, use the exterior power embedding into projective space and show that the image consisting of rank-one tensors is closed, and it represents the Grassmannian functor. This is in fact a closed immersion, see here. The idea is that if we put a -matrix containing a identity submatrix, then the set of -minor determines the remaining entries.)
For a commutative affine algebraic group , show that and are closed subgroups and there is an isomorphism . (The fact that they are subgroups follow from embedding into and uses the fact that commuting elements can be simultaneously diagonalized. The closedness of follows from being constructible. For more details see here. The fact that this is an isomorphism follows from bijectivity, see here)
Assume . Let be an element. Show that is unipotent iff for some and is semisimple iff for some (For unipotent this is easy; for semisimplicity note that every eigenvalue of lies in some finite field. This can be used to show Jordan decomposition in this case).