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.)