MATH32062
2 hours
THE UNIVERSITY OF MANCHESTER
INTRODUCTION TO ALGEBRAIC GEOMETRY
Date:
Time:
Answer THREE of the FOUR questions.
If more than THREE questions are attempted, then credit will be given for the
best THREE answers.
Electronic calculators are

FORMATION EVALUATION
Lecture 7
Lithologies
1
Importance of Lithology
Without a good Lithology Interpretation we
cannot calculate
the correct porosity
OR the correct hydrocarbon
saturation
2
Importance of Lithology
We need to know lithology to determine ma

Lecture 6
Formation density and neutron
logs
Formation Density Log
Formation density logs
These are active gamma logs, i.e., there is a source of
gamma rays on the tool as well as 2 detectors
The density logs use a complex compensation to
remove mudcake e

Relative Permeability
Diagrams:
A model reservoir
Paul Glover
1
Darcys Law: flow rate (laminar)
Q = kaA(DP/L)/m
Absolute permeability (ka) in ideal case, a property
of the rock only, not the fluid.
2
Flow and effective permeability (ke)
What if there is m

4
Projective space and projective varieties
One of the motivations for projective geometry is to understand perspective.
It is well-known phenomenon that parallel lines such as railway tracks appear to meet at a point, called the vanishing point. It was n

3
Tangent spaces, dimension and singularities
Let H An (K) be a hypersurface dened by the equation f (x1 , x2 , . . . , xn ) =
0, where f K[x1 , x2 , . . . , xn ], and and let P H. One of the possible ways
of dening that the line cfw_P + tv | t K (v K n \

2
2.1
Morphisms, co-ordinate rings, rational maps,
and function elds
Morphisms and co-ordinate rings
The obvious functions to consider on an ane algebraic variety V An (K)
are the polynomials. Two polynomials f , g dene the same function V K
if and only i

MATH32062
Problem Sheet 8
All questions are over C.
1. Show that any curve in A2 dened by an equation of the form
y 2 = x3 + ax2 + bx + c
is irreducible.
2. Let E be the elliptic curve given by the ane equation
2x3 + y 3 x2 x y = 0
Let (0, 0) be the zero

MATH32062
Problem Sheet 6
1. Homogenise the polynomial x3 + x2 y 2xy 2 x2 + xy y + 1 by using
x = X/Z and y = Y /Z. What are the points at innity (Z = 0) of the
projective variety dened by the resulting homogeneous polynomial?
2. Find projective transform

MATH32062
Problem Sheet 4
1. Let t be the co-ordinate on A1 (C) and let x, y be the co-ordinates on
A2 (C). Let C = V( y 3 x2 3xy + x + 3y ) A2 (C).
(a) Show that (t) = (t3 + 1, t2 + t) is a morphism A1 C.
(b) Show that is not an isomorphism A1 C. (Hint:

MATH32062
Problem Sheet 2
In questions 1, 2, 5 and 6, K is an arbitrary eld and V is a nite-dimensional
vector space over K, but you can assume K = R if you feel more comfortable
working with the real numbers.
1. Let U1 , U2 be ane subspaces of V . Prove

MATH32062
2 hours
THE UNIVERSITY OF MANCHESTER
INTRODUCTION TO ALGEBRAIC GEOMETRY
Date: 2 June, 2011
Time: 09:4511:45
Answer three of the four questions. If you answer all four questions, the three
best answers will be taken into account. All questions ca

MATH32062
2 hours
THE UNIVERSITY OF MANCHESTER
INTRODUCTION TO ALGEBRAIC GEOMETRY
Date: 1 June, 2012
Time: 9:4511:45
Answer three of the four questions. If you answer all four questions, the three
best answers will be taken into account. All questions car

MATH32062
Problem Sheet 3
1. (Most, if not all of this question was covered in MATH20212.) Let R be
a commutative ring and and let I be an ideal in R.
(a) Prove that I is maximal if and only if R/I is a eld.
(b) Prove that I is a prime ideal if and only i

FORMATION EVALUATION
Lecture 10
NMR, Image logs &
Final exam
Formation Evaluation. Session 10
1
Formation Evaluation. Session 10
2
Nuclear Magnetic Resonance
A new generation of tools meant NMR became
popular in the 1990s; similar technology to medical
M