NINTH SEMESTER THESIS

Algebraic Curves and Riemann-Roch Theorem

Submitted by:
Advaith R Nair
M0191303

in fulfillment of requirements
of 9th semester of integrated MSc

Project Advisor:
Prof. Omprokash Das
Department of Mathematics
TIFR, Mumbai
 Acknowledgements

I sincerely thank my project advisor Prof.Omprokash Das for helping me complete the project
fruitfully. I also thank the faculty members at UM DAE CEBS for giving me this amazing
opportunity to learn a lot of new things and also for their proper guidance and motivation in
the preceding semesters which has enabled me to undertake this task.
 NINTH SEMESTER THESIS 1

Contents
1. Algebraic Sets 2
2. Dimension of a variety 2
3. Intersection of Affine varieties 3
4. Bezout’s Theorem 4
5. Why morphisms? 5
6. Properties of Varieties 6
7. Rational Maps 7
8. Quadratic Transformation 8
9. Rational Functions on non-singular curves 9
10. Differentials on a Curve 11
11. Appendix 12
2 NINTH SEMESTER THESIS

1. Algebraic Sets
The space of n-tuples over a field k is called an affine n-space over k (An or An (k)). Let F ∈ k[x1 , x2 ..xn ].
P ∈ An (k) is a zero of F if F (P ) = 0 and the set of all such zeroes is called the hypersurface defined by
F (denoted as V(F)). A hypersurface defined by a degree 1 polynomial is called a hyperplane. One can also
define V (S) where S ⊆ k[x1 , x2 ...xn ] which is the common zeroes of all polynomials in S. Such a subset of the
affine space is called an affine algebraic set. We can associate an ideal to every such set because the affine
set for some T ⊆ k[x1 , x2 ...xn ] is the same as the one for the ideal I generated by it. Projective n-space
(Pnk ) over field k consists of equivalence classes of (n + 1)− tuples under the equivalence relation x ∼ λx, where
x is [x1 : x2 ..xn+1 ]. If every element in the equivalence class of x satisfies a polynomial then it satisfies every
homogenous polynomials (forms) of which it is a sum. Thus the algebraic sets of a projective space correspond
to homogenous ideals of k[x1 , x2 ...xn+1 ]. There is a natural one-to-one correspondence of points in Pn and
lines through origin in An+1 . One could also form mixed spaces where we take product of affine and projective
spaces. The algebraic set here are defined by polynomials homogenous in each set of variables corresponding to
the projective components. In any case one could see that complements of algebraic sets form a topology called
the Zariski topology. Pn can be written as union of Ui = {[x1 : x2 .. : xn+1 ] ∈ Pnk |xi ̸= 0} each of which is
homeomorphic to An . A subspace of a topological space is said to be irreducible if it can’t be written as a
union of its proper closed subsets. If k is algebraically closed, for affine spaces we have a unique radical ideal
corresponding to a given algebraic set. This is precisely the Hilbert’s Nullstellensatz. Combining this with the
fact that the polynomial ring is noetherian one deduces that any non-empty subfamily of a family of closed
subsets of such a space has a minimal element. A topological space with this property is said to be noetherian.
Theorem 1.1. A closed subset of a noetherian space can be written uniquely as a finite union of irreducible
closed subsets none containing the other.
Proof. Let G be the class of non-empty closed subsets which can’t be written as a finite union of irreducible
closed subsets. If G is non-empty, the minimal element (which is not irreducible) can be written as a union of
algebraic subsets Y and Y ′ both of which can be written as a finite union of irreducible closed subsets thus giving
us a contradiction. Now to prove the uniqueness of a given representation consider two such representations
r r′ r
Yj′ of a closed set Y. Y1′ = (Y1′ ∩ Yi ) so Y1′ ⊆ Yi for some i, say i = 1. Similarly Y1 ⊆ Yj′ for
S S S
Yi and
i=1 j=1 i=1
some j. This implies j = 1 and thus Y1 = Y1′ . Similarly by induction the uniqueness is also proved. □
Example 1.2. Let Y be the algebraic set in A3 defined by x2 − yz and xz − x. Then Y has the irreducible
components given by V ((x, y)); V ((x, z)) and V ((x2 − yz, z − 1)). The third is irreducible since it is isomorphic
to the variety V (x2 − y) in A2 which will be made clear later.
This fact helps one to reduce a problem involving algebraic sets to the irreducible algebraic sets which makes
them important in this study. From now on they will be called varieties. An ideal corresponding to a variety
is prime, so for a variety V one could define a corresponding integral domain Γ(V ) = k[x1 , x2 ...xn ]/I(V ) called
the co-ordinate ring of V. For varieties in affine space, It sits inside the ring of k-valued functions on V and
is same as the set of polynomial functions on V, but in non-affine cases, these are clearly not functions. Define
the quotient field of Γ(V ) to be the function field of V (k(V )). Elements of these sets define a function on
a subset of V these are called the rational functions of V. A rational function h of V is said to be defined
at P ∈ V if for some f,g ∈ k[x1 , x2 , ...xn ], h = f /g at P and g(P ) ̸= 0. All such functions form a subring of
k(V ) called OP (V ). OP (V ) is the localisation of the polynomial ring at the maximal ideal corresponding to P.
This can be seen easily for an affine variety, the proof for the general case is found in the appendix. The set of
points on V where a rational function is not defined is called its pole set. Pole set of f ∈ k(V ) is closed in V,
since Pole set = V (Jf ) where Jf = {G ∈ k[x1 x2 ...xn ]|Ḡf ∈ Γ(V )}.
Example 1.3. Let V = V (Y 2 − X 2 (X + 1). The pole set of z = Y /X is V ((X, Y )) while the pole set of z 2 is
empty. If for an affine variety V if Γ(V ) is a UFD, then for any representation of z, P will be in its pole set if
it is the zero of the polynomial in the denominator, it is clear from this example that one can’t say the same is
true for other cases.

2. Dimension of a variety
An affine variety V can be given a topological dimension which is the length of the largest descending chain
of irreducible closed sets starting at V. This translates directly into the Krull dimension of its co-ordinate ring.
It is interesting to note that the height of a prime ideal p of the co-ordinate ring is the same as the dimension
of the coordinate ring localised at p. Noether normalisation lemma states that there exists x1 , x2 , x3 ..xd in
Γ(V ) such that they are algebraically independent over k and Γ(V ) is integral over A = k[x1 , x2 , ...xd ] (Implying
they have the same Krull dimension d). Which means K is algebraic over the quotient field of A say Q(A),
and trk K = trk Q(A) = d. Hence Dimension of a variety is same as the transcendence degree of its
 NINTH SEMESTER THESIS 3

function field over k. Function field being a set of functions over subsets of V, dimV also gives information
about geometry of a variety.
Theorem 2.1. The affine varieties of dimension (n − 1) of An corresponds to hypersurfaces of irreducible
polynomials.
Proof. Given a variety of V (f ) where f ∈ k[x1 , x2 ..xn ] is irreducible, take the localisation of the co-ordinate
ring at the prime ideal (f ). As a consequence of dimension theorem for local noetherian rings, one has htp to
be 1. Conversely let htp = 1, and p = (f1 , f2 ...fk ) where each fi is irreducible and k is the minimum number of
generators of p, then p ⊇ (fi ) ⊇ 0 which violates the height condition unless k = 1. □
Remark 2.2. V (I) = ϕ in affine case iff ideal I is the polynomial ring. It is not the same in other cases, for
instance in Pn it can happen if the zero set of the ideal is (0, 0...(n+1)−times) which corresponds to ideals whose
radical is (X1 , X2 , X3 ...Xn+1 ). This doesn’t change the duality of radical ideals and varieties. For example in
this very case we can’t say that k[x1 , x2 ...xn ] is also an ’allowable’ radical ideal corresponding to the empty
set since it is not a homogenous ideal. For this reason the theorem about decomposition of algebraic sets into
varieties work for all kinds of spaces (mixed,affine or projective).
Example 2.3. A subvariety of An is called linear if the ideal corresponding to it is generated by only linear
polynomials say (f1 , f2 ..fr ). Dimension of this variety is same as the size of maximal linearly independent subset
of fi s after a coordinate change such that constant terms of all fi are 0 (using results in 3).

3. Intersection of Affine varieties

Given varieties Y and Z in an affine spaceAn one wishes to study their intersection. Let dim(Y ) = r and
dim(Z) = s, Y ∩ Z is isomorphic to variety (Y × Z) ∩ ∆. dim(Y × Z) = r + s. Since ∆ is an intersection of
n hypersurfaces one is tempted to work out the case for intersection of a variety with a hypersurface. Say X is
a variety of dimension r′ and X’ is a hypersurface not containing X. Then the ideal corresponding to X ∩ X ′
is the prime ideal (f ) which has height = 1 by Krull’s Hauptidealsatz and thus X ∩ X ′ has dimension r − 1.
Thus the dimension of a variety remains unchanged or reduces by 1 when intersected by a hypersurface. So the
dimension of dim(Y ∩ Z) ≥ r + s − n. Thus as long as r + s − n > 0 the intersection of varieties is non-empty.
Given any two curves (varieties of dimension 1) with no common component, their intersection is a finite set
of points. For this reason the ring obtained by quotienting the polynomial ring with the ideal corresponding
to the intersection of the varieties is an artinian ring which due to the structure theorem is a product of local
rings at each of these points. These local rings contains information about the nature of intersection at their
respective points. The dimension of the local ring (as a vector space over k) corresponding to an intersection
point is called the intersection number of that point denoted I(P, F ∩ G). On a plane curve C defined by an
equation F (X, Y ) = 0 one notes that the form of the lowest degree can be written as product of linear factors
and they correspond to the tangents at point P = (0, 0, ..0) of the curve. The degree of this form is called the
multiplicity of P (mP (C)). If the multiplicity of P is greater than 1, it is called a multiple point if all the factors
of the minimum form are distinct then P is called an ordinary multiple point. Any irreducible plane curve has
only a finite number of multiple points. This is because they lie in the intersection of irreducible curves with
no common components (F, ∂F ∂F
∂x and ∂y ).

Example 3.1. In curve Y 2 = X 3 the lowest homogenous component can be factorised as Y · Y , implying that
the point P = (0, 0) is a multiple point on the curve but not an ordinary multiple point. At the same time
take for example the curve Y 2 = X 2 (X + 1), The factorisation in this case is (Y + X)(Y − X) which indicates
multiple point at P but this time this is an ordinary one. The intuition about the tangents can be seen to be true
from the diagrams below
4 NINTH SEMESTER THESIS

Theorem 3.2. A points on the curve C = V (F ) are simple (mP (C) = 1) if and only if OP (F ) is a DVR.
Proof. Let P be a simple point on C. After appropriate co-ordinate changes, assume Y = 0 tangent to C at
P = (0, 0, ...0) and X = 0 to be not. Then F = Y G − X 2 H where G is a polynomial with non zero constant
term. Thus in OP (F ), y = x2 hg −1 thus mP (F ) = (x). Before proving the converse one must note that
O/mn has a finite dimension over k since it is an Artin ring. Also from the structure theorem of Artin rings
k[x, y]/(I n , F ) ∼
= OP (A2 )/(I n , F ) ∼
= O/mn for n > m = mP (F ) where I = (x, y). One can use the following
exact sequence to calculate the dimension of O/mn :
ψ ϕ
0 → k[x, y]/I n−m −
→ k[x, y]/I n −
→ k[x, y]/(I n , F ) → 0
ψ is the multiplicaltion by F and ϕ is the canonical map. dimk k[x, y]/I r = r(r+1)/2 thus dimk k[x, y]/(I n , F ) =
mn − m(m−1)
2 which implies mP (C) = dimk mn /mn+1 . Now if O is a DVR, one has the desired result. □
Theorem 3.3. I(P, F ∩ G) ≥ mP (F )mP (G)
I(P, F ∩ GH) = I(P, F ∩ G) + I(P, F ∩ H)
Proof. The following diagram is easily seen to be exact.(mp (F ) = m,mP (G) = n)
ψ
k[x, y]/I n × k[x, y]/I m −
→ k[x, y]/I n+m → k[x, y]/(I m+n , F, G) → 0
where ψ(A, B) = AF + BG.
I(P, F ∩ G) = dimk O/(F, G) ≥ dimk O/(I m+n , F, G) = dimk k[x, y]/(I m+n , F, G)
≥ dimk k[x, y]/I n+m − dimk k[x, y]/I n − dimk k[x, y]/I m (from the exactness of the diagram)
= mn
To prove the second statement (for finite case), consider the diagram
ψ
0 → O/(F, H) −
→ O/(F, GH) → O/(F, G) → 0
where ψ(z) = Gz. Say Gz = uF + vGH then there exists polynomials A = Su, B = Sv, C = Sz, where S is
also a polynomial such that G(C − BH) = AF which implies z = C/S = (D/S)F + (B/S)H. Thus the above
diagram is exact, hence proved. □
Theorem 3.4. I(P, F ∩ G) = mP (F )mP (G) iff F and G doesn’t have any common tangents at P.
Proof. I(P, F ∩ G) = mP (F )mP (G) holds if and only if the inequalities appearing in the proof of the first part
of previous theorem are equalities which happens if and only if the following are true
i) I t ⊂ (F, G)O for t ≥ m + n
ii) ψ is injective
Suppose (ii) holds, then ψ(Ā, B̄) = 0 this means the factors of the minimal forms are distinct since otherwise the
pair of residues of the product of non-common terms in them which are non-zero in k[x, y]/I m and k[x, y]/I n
will be in kernel of ψ conversely if the tangents are distinct then ψ(Ā, B̄) = 0 implies Ar |Gn and Bs |Fm . So ψ
is injective if and only if the tangents of F and G are distinct. As for the second part one already knows that
as a consequence of nullstellensatz that, I t sits in (F, G) for sufficiently large t. Also if the tangents of F and
G say {L1 , L2 ...Lm } and {M1 , M2 ...Mn } are distinct then {Aij = L1 L2 ...Li M1 M2 ...Mj |i + j = d} (Li = Lm
and Mj = Mn for i > m and j > n repectively) form basis of d-forms in k[x, y] af a vector spacre over k. If
i + j > m + n − 1 either i ≥ m or g ≥ n say i ≥ m. Then Aij = Am0 B where B is some form of degree i + j − m,
thus Aij = B(F − F ′ ). Since BF and BF ′ are of degree greater than i + j + 1. We can continue doing this
for larger and larger values of i+j until I i+j ⊂ (F, G)O. Therefore Aij ∈ (F, G)O whenever i + j > m + n − 1.
Hence the theorem is proved. □

4. Bezout’s Theorem
Given any two varieties of dimensions r and s in Pn , if r + s ≥ n then their respective cones with dimensions
r + 1 and s + 1 intersect in An+1 such that the irreducible components of their intersection has dimension
≥ (r + 1) + (s + 1) − (n + 1) > 0 which implies their corresponding projective varieties have a non-empty
intersection (which contains point other than the origin). This means unlike in the case of affine plane any two
curves on a projective plane have non-empty intersection. Of course the number of points of intersection will
be finite as all of them lie on intersections of finitely many pairs. As mentioned in the appendix, one can define
I(P, F ∩ G) = dimk (OP (A2 )/(F∗ , G∗ )) like OP (V ) was defined to be OP (V∗ ). The proof of this fact is omitted.
Theorem
P 4.1. F and G are projective plane curves of degree m and n respectively (no common component).
then I(P, F ∩ G) = mn
P ∈P2

Proof. One can assume that all the intersection points lie on points where Z ̸= 0, in which case the LHS by the
property of Artin rings, becomes dimk k[x, y]/(F∗ , G∗ ). Observe that the map
z
k[x, y, z]/(F, G) −
→ k[x, y, z]/(F, G)
 NINTH SEMESTER THESIS 5

is injective. Let Γ = k[x, y, z]/(F, G) and R = k[x, y, z], then the following diagram is exact:
ψ ϕ
0→R−
→ R×R−
→R→Γ→0
where ψ(C) = (GC, −F C) and ϕ(A, B) = AF + BG from here one concludes that dim(Γd ) = mn for d ≥ m + n.
say Γd has basis {A1 , A2 ...Amn } then {z r A1 , z r A2 ...z r Amn } is the basis of Γd+r . If (Ai )∗ = ai then their
residues with respect to (F∗ , G∗ ) are linearly independent and generate k[x, y]/(F∗ , G∗ ). □

Due to this reason, the non-singular projective plane curves are irreducible since otherwise the points of
intersection of their irreducible components will be singular.

5. Why morphisms?
Till now, only the algebraically irreducible sets in affine and projective spaces have been discussed. Mul-
tispaces are spaces of the form X = Pn1 × Pn2 × ...Pnr × Am . The algebraic sets here correspond to ideals
in k[x10 , x12 , ...xrnr , x1 , x2 ..xm ], polynomials of which are homogenous in each set Si = {xij |0 ≤ j ≤ ni },
1 ≤ i ≤ r. One sees that there exists open sets in Pn which are homeomorphic to An thus making the affine
varieties in An homeomorphic to some quasi-projective varieties in Pn , which leads to the question whether
something similar can be done in the multispace case. Before answering this one needs to extend the definition
of varieties. From now on a variety is an open subset of an irreducible algebraic set of a multispace. An open
(resp. closed) subvariety of a variety is an open (resp.closed) subset of a variety and also a variety on its own.
One can thus also extend the definition of Γ(U ) to be the set of rational functions defined on U. The Γ(U ) is
same as the coordinate ring of U when U ∈ An is an affine variety. This can be proved as follows (Here the
co-ordinate ring is denoted by A(U)):

Note that A(U )mP =∼ OP (U )

T T
now, A(U ) ⊆ Γ(U ) ⊆ A(U )mP since Γ(U ) = OP (U )
P ∈U T P ∈U
Since A(U) is an integral domain, A(U ) = A(U )mP from which the desired result follows.
P ∈U
When one wants to say that some variety V in a space X is same as another variety V’ in a space X’ one
should also demand for them to have isomorphic rings of their global rational functions since one wishes to
study the geometric properties of a variety using its algebraic attributes, but does a homeomorphism guarantee
that? Here is a counterexample: ϕ : A1 → A2 is an homeomorphism of A1 with the curve given by the equation
y 2 = x3 . But their rings of global regular functions are k[t] and k[t2 , t3 ] respectively, which are not isomorphic.
Here comes the need for introducing a stronger notion than homeomorphisms.

Definition 5.1. X and Y be varieties, a continuous map ϕ : X → Y is said to be a morphism if, given f ∈ Γ(U ),
U an open subset of Y, f ∈ Γ(ϕ−1 (U )).
A morphism which has an inverse which is also a morphism is called an isomorphism. An isomorphism
is easily seen to be a homeomorphism. It is also clear that for any homeomorphism from V to V’ to be an
isomorphism, one requires that for any given open subset U ⊆ V homeomorphic to open subset U ′ ⊆ V ′ ,
Γ(U ) ∼= Γ(U ′ ). One can immediately conclude from this observation that the open subsets of a projective
n-space homeomorphic to An is indeed isomorphic to it. The following theorem gives one the morphisms from
an affine variety to another.

Theorem 5.2. X ⊆ Am ,Y ⊆ An are affine varieties, There is a one-to-one correspondence between morphisms
ϕ : X → Y and k-algebra homomorphisms from Γ(Y ) to Γ(X).
Proof. Suppose ϕ : X → Y is a morphism. then xi ◦ ϕ is a global rational function on X for 1 ≤ i ≤ n by the
definition of a morphism. which is same as saying that xi ◦ ϕ is a polynomial function on X since X is an affine
variety. Conversely if ϕ is a polynomial map, it is clearly continuous and for a given global rational function f
on an open set U ⊆ Y , f ◦ ϕ is a global rational function on ϕ−1 (U ) since locally f is a quotient of polynomials.
It is then easily seen that the set polynomial maps from X to Y are precisely the set of homomorphisms between
them, hence proved. □

Example 5.3. V ⊆ An is an affine variety, f ∈ k[x1 , x2 , . . . , xn ], then

V ∗ = {(x1 , x2 , . . . , xn , f (x1 , x2 ..xn )) | (x1 , x2 , . . . , xn ) ∈ V } ⊆ An+1
is isomorphic to V. This is because the projection from V ∗ to V, π(x1 , x2 ..xn+1 ) = (x1 , x2 ..xn ) is clearly a
homeomorphism with inverse ψ(x1 , x2 ..xn ) = (x1 , x2 ..xn , f (x1 , x2 ..xn )). Moroever by previous theorem, both of
them are morphisms, hence proved.
6 NINTH SEMESTER THESIS

Example 5.4. f ∈ k[x1 , x2 ..xn ] and Vf = {(x1 , x2 ..xn )|f (x1 , x2 ..xn ) ̸= 0}. Vf is isomorphic to an affine
variety in An+1 . Let V ′ = {(x1 , x2 ..xn , xn+1 )|xn+1 f (x1 , x2 ..xn ) − 1 = 0}. The projection from V’ to its first n
co-ordinates define a homemorphism of these sets. Now using the proof of the previous theorem the projection
is also a morphism. Consider the inverse of the projection map π −1 . Let s ∈ Γ(U ′ ), U ′ ⊆ V ′ . Locally
s = p/q where p, q ∈ Γ(V ′ ). (s ◦ π −1 )(α) = P (π −1 (α))/Q(π −1 (α)) where P, Q ∈ k[x1 , x2 ..xn ], P = p, Q = q,
α ∈ Vf . s is defined at every point in U ′ also there exists N ∈ N such that f N · (p ◦ π −1 )(α) = P ′ (α) and
f N · (q ◦ π −1 )(α) = Q′ (α) P ′ , Q′ ∈ k[x1 , x2 ..xn ]. Thus (s ◦ π −1 ) is a regular function in π(U ′ ) and hence π −1
is also a morphism.

6. Properties of Varieties
This section gives some insights into the structure of a given variety in terms of projective and affine vari-
eties. The most important tool one requires is the map S : Pn × Pm → Pn+m+mn given by S([x0 : x1 ..xn ], [y0 :
y1 ..ym ]) = [x0 y0 : x0 y1 ..xn ym ]. S(Pn × Pm ) = V where V = V ({Tij Tkl − Til Tkj |0 ≤ i, k ≤ n; 0 ≤ j, l ≤ m})
where {Tij |0 ≤ i ≤ n; 0 ≤ j ≤ m} are co-ordinates of Pn+m+mn . This map is called the Segre embedding.

We make a simple but useful remark about isomorphism of varieties.

Remark 6.1. To prove that this is an isomorphism one should note the following: If f : X → Y such that there
are families of open subsets {Uα }α∈A and {Vα }α∈A of X and Y respectively and f (Uα ) ⊂ Vα , f is a morphism
iff the restriction to each Uα is a morphism.
Theorem 6.2. Segre embedding is an isomorphism from Pn × Pm to the projective variety V ⊆ Pn+m+mn .
Proof. The map is clearly surjective, one should prove it is injective. let S(α, β) = S(α′ , β ′ ) = γ. αi βj = λαi′ βj′
for 0 ≤ i ≤ n, 0 ≤ j ≤ m and fome fixed 0 ̸= λ ∈ k (Call these equalities ∗(i, j)). For some fixed i0 and j0 ,
αi0 ̸= 0 βj0 ̸= 0. Using ∗(i0 , j) one can deduce that βj = λ′ βj′ = 0 for all j and some fixed 0 ̸= λ′ ∈ k, thus
β = β ′ and similarly using ∗(i, j0 ) one can deduce α = α′ .
Using the remark above, one may just prove that S : Ui × Uj → V ∩ Uij is an isomorphism, but since
these are affine varieties one just need to show that the induced homomorphism between the respective co-
ordinate rings is an isomorphism (using theorem 3.1.1). Without loss of generality take i=j=0. Γ(U0 ×
U0 ) = k[x1 , x2 ..xn , y1 , y2 ..ym ] and Γ(V ∩ U00 ) = k[T01 ..]/I(V ). Where V is the algebraic set corresponding to
{GTkl −Til Tkj |G = 1 when i = j = 0, G = Tij otherwise}. But the homomorphism ϕ : Γ(U0 ×U0 ) → Γ(V ∩U00 )
such that ϕ(xi ) = Ti0 and ϕ(yj ) = T0j is an isomorphism proving that S −1 (Since S is a homeomorphism) is an
isomorphism. □

The above theorem can be extended to any space of the form Pn1 × Pn2 × ...Pnr to show that it is isomorphic
to a variety in a projective space PN for some N. The above theorem could also be extended to any closed
subvariety of such a space once we prove the same for a closed subvariety W in the base case replacing Pn × Pm
by W and V by an appropriate projective variety VW ⊆ V . Thus any variety is an open subvariety of
some projective variety. Using this theorem given any variety X and a point P in X, one can find some
neighbourhood X’ which is isomorphic to some open subvariety X” of An . Note that for any open neighbourhood
U of the variety X, we can find a polynomial F in k[x1 , x2 ..xn ] such that for the point P’ in X” corresponding
to P and all Q’ corresponding to points Q in X’ U such that F (P ′ ) ̸= 0 and F (Q′ ) = 0. Using exercise
3.1.2 one could easily see that Xf′′ where f is the residue of F in Γ(X ′ ) is an affine variety. Henceforth such a
neighbourhood will be referred to as an affine neighbourhood. Thus Affine neighbourhoods form a basis
of Zariski topology of any given variety. The following theorem enables one to define some new morphisms
from the existing ones.
Theorem 6.3. A,B be multispaces; V ⊆ A, W ⊆ B be closed subvarieties then V × W is a closed subvariety
of A × B
Proof. Let V × W = Z1 ∪ Z2 . One may define Ui = {y ∈ W |V × {y} ⊈ Zi }, Since V × {y} is irreducible
U1 ∩ U2 = ϕ. WLOG pick β ∈ U1 . There exists a defining multiform of Z1 say F(x,y) and α ∈ V such
that F (α, β) ̸= 0. Define the multiform G(y) = F (α, y) in W. {y ∈ W |G(y) ̸= 0} is an affine open subset of
β contained in U1 . Thus U1 and similarly U2 are open which implies one of them has to be empty (due to
irreducibility of W and V × {y}). WLOG suppose U1 = ϕ then V × W ⊂ Z1 .
□

Remark 6.4. One immediately sees that following maps are morphisms.
a) Projection map π : X × Y → X.
b) (f, g) : Z → X × Y when f : Z → X and g : Z → Y are given to be morphisms.
c) f × g : X × X ′ → Y × Y ′ when f : X → X ′ and g : Y → Y ′ are given to be morphisms
d) δX : X → ∆X = {(x, x)|x ∈ X} ⊂ X × X
 NINTH SEMESTER THESIS 7

Example 6.5. Consider the map ρd : Pn → PN where N = (n+d n ) − 1 by ρd (P ) = [M0 (P ), M1 (P ), ..., MN (P )]

where Mi are monomials of degree d in variables {y0 , y1 , ..., yn }. This is a morphism in every affine patch thus
a morphism of projective spaces. Moreover since this map is injective this is an embedding of Pn into PN .
6.1. Image of a projective variety under a morphism. Consider the map π : Pn × Am → Am which is
the projection map onto Am . The space Pn × Am has the appropriate Zariski topology induced by zero sets of
polynomials in k[x0 , x2 , ..., xn , y1 , y2 , ..., ym ] homogenous in {x0 , x1 , ..., xn }. They correspond to ideals of R =
∞
Sd ⊗k k[y1 , y2 , ..., ym ]., where Sd is homogenous polynomials of degree d in k[x0 , x1 , ..., xn ]. Let Z ⊂ Pn × Am
L
d=0
be a closed subset of Pn × Am . Let b ∈ Am be such that b ∈ / π(Z), then π −1 (b) = Pn × {b} = V (R · mb ) where mb
is the maximal ideal in k[y1 , y2 , ..., ym ] corresponding to b. If I is such that Z = V (I), one has V (I) ∩ R · mb = ϕ.
By nullstellensatz, Rd · mb + Id = Rd for all sufficiently large d. By Nakayama’s lemma there exists f ∈ mb
such that (1 + f )Rd ⊆ Id for all large d implying that (1 + f ) ∈ I0 and thus b ∈ / V (I0 ). Conversely whenever
b ∈ V (I0 ), b ∈ π(Z) which means π(Z) = V (I0 ) concluding that π is a closed map. Using this one can deduce
that for any projective variety Y and some variety X, the projection map of π : X ×Y → X is a closed map. Now
consider a morphism of varieties Φ : X → Y where X is a projective variety. The map (Φ, id) : X × Y → Y × Y .
∆ = {(y, y)|y ∈ Y } is a closed subset of Y × Y , thus (Φ, id)−1 (∆) = {(x, y)|Φ(x) = y} is a closed set of X × Y .
The image of X under Φ is same as the image of (Φ, id)−1 (∆) under π which is closed in Y since π is closed.

7. Rational Maps
It was shown that isomorphic affine varieties have isomorphic co-ordinate rings, however this fails in the case
of general varieties which can be shown by the embedding v : P1 → P3 ; v([x0 : x1 ]) = [x30 : x20 x1 : x0 x21 : x31 ].
Eventhough one has isomorphic rational function rings, it is not very useful in many instances. For example
the ring of rational functions for any given projective variety is the base field k. However for a given variety one
can find an open subset inside it which is affine, so what if for any two given varieties one has these embedded
open affine subvarieties which are isomorphic? Then one would have their corresponding function fields also to
be isomorphic! Thus motivated from this fact, one could define for any two given varieties X and Y, a rational
map f from X to Y which is a morphism on an open subvariety U ⊆ X called the domain of the rational
map to Y such that it can’t be extended to any larger open subset of X. When the image of the map is dense
in Y, it is called a dominating rational map. fα , the restriction of f to some open set Uα ⊆ U is called
a representation of the rational map. Given a representation f : U → V of a dominating rational map F
from X to Y, one gets a homomorphism f˜ : Γ(V ) → Γ(U ). One sees that images of any two given elements
in Γ(V ) coincide if and only if they coincide on f(U). But since the representations are morphisms, Using (b)
and (d) of 6.4 one sees that they coincide on a closed set, thus f˜ is injective (since f(U) is dense in V), and
can be extended uniquely to an injective homomorphism F̃ : k(Y ) → k(X). Conversely given an injective
homomorphism ϕ : k(Y ) → k(X), assume them to be affine varieties (basically their affine open subvarieties).
Then if Γ(Y ) = k[y1 , y2 ..yn ] and ϕ(yi ) = ai /bi ; ai ,bi ∈ Γ(X) then ϕ(Γ(Y )) ⊂ Γ(Xb ) where b = b1 b2 b3 ..bn . Since
ϕ restricted to Γ(Y ) is a homomorphism of coordinate rings of affine varieties it is induced by the respective
morphism say f : Xb → Y whose image is dense in Y since f˜ = ϕ is injective. One could show that for a
rational map which induces an isomorphism of function fields, there is a representation morphism which is an
isomorphism. Such a map is called birational map and the varieties in this case are said to be birationally
equivalent. One immediately notes that any n-dimensional variety is birationally equivalent to a hypersurface
in An or Pn . Also note that Pn+m is birationally equivalent to Pn × Pm .
Example 7.1. The map ϕ : A1 → V (X 2 − Y 3 , Y 2 − Z 3 ) given by ϕ(t) = (t9 , t6 , t4 ) is a birational map since
the map is an injective morphism into A3 with image of dimension 1 lying in V which also has dimension 1.
Any rational map between curves say C and C ′ is either dominating or a constant. This can be shown as
follows: If the map is not dense, closure of f (U ) will be a proper closed subvariety of C ′ say W. Let p be the
prime ideal of Γ(C ′ ) corresponding to W. K(C ′ ) ⊃ Γ(C ′ ) ⊃ k, and trk K(C ′ ) = 1. So there exists polynomial F
for any given x ∈ Γ(W ) and y ∈ p such that F (x, y) = a0 (x) + a1 (x)y + a2 (x)y 2 ... = 0. Γ(C ′ )/p is a field lying
between k and K(C ′ ) but a0 (X) ̸= 0 but a0 (x̄) = 0 so Γ(C ′ )/p = k. Which implies f (U ) ⊆ W = {Q} for some
Q ∈ C ′.
7.1. Domain of a rational map on a curve. Let f : X → Y be a rational map of varieties X and Y . Note
that whenever f (P ) = Q then OP (X) dominates f˜(OQ (Y )). Conversely if OP (X) dominates f˜(OQ (Y )), then
for some affine neighbourhoods V of P and W of Q, one can find b ∈ Γ(V ), b(P ) ̸= 0 such that f˜(Γ(W )) ⊂ Γ(Vb )
and by 5.2, there exists a morphism f : Vb → W such that f˜ is induced by f . Therefore if g ∈ Γ(W ), g(Q) = 0
implies g ◦ f vanishes at P thus f (P ) = Q. Let X = C and Y = C ′ be curves and f be a dominant. One wishes
to identify the points on C such that OP (C) dominates OQ (C ′ ) for some Q in C ′ . Let’s say we got an affine
open subset V of C’ such that OP (C) ⊃ Γ(V ). Then if J = mP (C) ∩ Γ(V ), there is a closed subvariety W of V
corresponding to J which is either a singleton say W = Q in which case OP (C) dominates OQ (C ′ ); or W = V .
8 NINTH SEMESTER THESIS

In the latter case k(C) ⊃ OP (C) ⊃ k(C ′ ). Since k(C’) is a finite algebraic extension of k(C), OP (C) has to be
a field. Such a Q if it exists for a given P ∈ C is unique, the trouble arises at two points a) Existence of the
aforementioned affine neighbourhood b) OP (C) being a field. If OP (C) is a DVR we can get rid of both the
issues. C can be considered as a closed subvariety of Pn such that it has non-empty intersection with each Ui
and Γh (C) = k[X1 , X2 ...Xn+1 ]/I(C) = k[x1 , x2 ...xn+1 ] and max{ord(xi /xj )|1 ≤ i, j ≤ n + 1} = ord(xi /xn+1 )
for some i. Then C∗ = C ∩ Un+1 is easily seen to satisfy condition (a) and also prevents condition (b) from
occuring. Thus the domain of a rational map on a curve has all the points on it whose local ring is
a DVR. The curve whose every point satisfies this condition is said to be nonsingular. Isomorphism classes
and birational equivalence classes of nonsingular curves coincide.
7.2. Blowup. Blowup of An at O = (0, 0, 0..., 0) is defined to be the closed subset X of An × Pn−1 given
by equations {xi yj = xj yi |i, j = 1, 2, ..n}. The projection map of X onto An say π is a morphism (6.4) but
when restricted to X − π −1 (O) is an isomorphism making X birationally equivalent to An . X is indeed a closed
subvariety since any point in π −1 (O) lies in closure of X − π −1 (O) which is irreducible (isomorphic to An − {O})
and open (π is a morphism) in X. This can be seen as follows: Say P = (O, [a1 : a2 : a3 ... : an ]) then consider
the line L in An given by {(x1 , x2 ..xn )|xi = ai t, t ∈ k}. π −1 (L − {O}) is given by xi = ai t and yi = ai the
closure of which certainly contains P. Blowup of a variety Y of An is the closure of π −1 (Y − {O}) which also
by the same reasoning is a variety. X can be written as a union of two open subvarieties Xi corresponding to
each Ui which are isomorphic to closed varieties in A3 given by Y = XZ (resp. X = ZY ) which are actually
isomorphic to A2 the isomorphism being ϕ(x, z) = (x, zx, z). But considering the local affine subvarieties, one
has to drop the relevant points in A2 to get a new subvariety of A2 isomorphic to Xi which turns out to be
either X = 0 or Y = 0. Thus we have that each Xi is birationally equivalent to A2 − {X = 0orY = 0} via
the projection map. Thus we deduce that ψ : A2 → A2 , ψ(x, z) = (x, xz) is a birational map from A2 to itself
whose domain is isomorphic to Xi . Using this map one could understand how the blowup of a curve C at O
(Closure of π −1 (C)) looks at one of its affine patches. The points corresponding to O on the blowup of curves
correspond uniquely to the distinct tangents at the point (except X = 0 or Y = 0 depending on which patch one
is considering). Example: Consider Y 2 = X 2 (X + 1). In the blowup corresponding to U2 , X 2 Z 2 = X 3 + X 2 .
Since one knows that blowup has to be irreducible it is given by z 2 = x+1 the points corresponding to the origin
are (0,1) and (0,-1) which correspond to projective co-ordinates of points in P2 [1:-1] and [1:1] representing the
tangents Y + X and Y − X. The multiplicities of points corresponding to O in the blowup is always bounded by
the intersection number of the curve with the X-axis or Y-axis (depending in the patch) which is equal to the
multiplicity of the tangent in the minimal form of the curve. thus if O is an ordinary multiple point, then all
the points corresponding to it in the blowup are simple. In this example z = 0 is the only tangents at (0,1) and
(0,-1). For the point O we have an affine neighbourhood W such that its preimage W ′ is an affine subset and
Γ(W ′ ) is generated finitely over Γ(W ) as a module and there exists x ∈ Γ(W ) such that it is xr−1 Γ(W ′ ) ⊂ (W ).
One can also define blowup on a projective space (pivoted on several points) such that it looks exactly like the
affine blowup in each of its affine patches where the pivots lie.

Figure 2. Blowup of Affine plane at origin

8. Quadratic Transformation
From the above section one gets a recipe to create a nonsingular curve birationally equivalent to a planar
projective curve. If we have such a curve birationally equivalent to some planar curve, then for any curve we get
a recipe for getting a nonsingular curve birationally equivalent to it (Since for any curve we have a planar curve
birationally equivalent to it see 7) which is what is described in this section. The Quadratic transformation
on a projective plane is given by: Q([x : y : z]) = [yz : xz : xy]. As one can see that this is not defined on
P = [0 : 0 : 1], P ′ = [0 : 1 : 0] and P ′′ = [1 : 0 : 0]. Q is a morphism (on P2 − {P, P ′ , P ′′ }. Since its image is
U ∪{P ′ , P, P ′′ } (where U = P2 −V (XY Z)) this gives an isomorphism from U to itself since Q = Q−1 on U. Since
Q is a birational map on P2 one gets for an irreducible curve in P2 a birationally equivalent curve C’ (closure of
 NINTH SEMESTER THESIS 9

Q−1 (C ∩ U ) which is also irreducible. All the points in that curve satisy F Q = F ([yz : xz : xy]) = Z r Y r ′ X r ′′ F ′
where r = mP (C), r′ = mP ′ (C) and r′′ = mP ′′ (C) and X,Y,Z doesn’t divide F ′ . Sine C’ is irreducible, it is
given by F ′ of degree 2n − r − r′ − r′′ . From the formula of F ′ one also deduces that mP (C ′ ) = n − r′ − r′′ ,
mP ′ (C ′ ) = n − r − r′′ and mP ′′ (C ′ ) = n − r′ − r. This computation leads to the observation that if none of
L = V (Z), L′ = V (Y ) or L′′ = V (X) are tangent to C at their corresponding fundamental points, the same
holds for C ′ also. Such a curve is said to be in good position (any curve can be brought to one by co-ordinate
changes). If further one has that L intersects C (transversally) in n distinct non-fundamental points L′ and L′′
intersect C in n − r non-fundamental points (transversally) each then the curve is said to be in an excellent
position.
Remark 8.1. For C,F,Q as above consider C∗ = V (F∗ = F (X, Y, 1)). Consider F∗′ with respect to the patch
where one of the points corresponding to P of C ′ lies. Then the map Q restricted to C∗′ to C∗ looks like partial
blowup map f (x, z) = (x, zx). Thus we have the same property of blowup map for the quadratic transformation
regarding existence of a local affine neighbourhood W of P and W ′ = Q−1 (W ) such that the Γ(W ′ ) is module
finite over Γ(W ) and existence of a t ∈ Γ(W ) such that tΓ(W ′ ) ⊂ Γ(W ).
Theorem 8.2. If C is in excellent position, C’ has the following properties:
(a) Multiplicities of points in C ∩ U and ordinary multiple points are preserved and lie in C ′ ∩ U .
(b) P , P ′ and P ′′ are ordinary multiple points in C ′ with multiplicity n, n − r, and n − r.
(c) There are no non-fundamental points on C ′ ∩ L′ or C ′ ∩ L′′
Proof. (a) is a consequence of Q being an isomorphism on U. Let C be a curve in good position, Write F ′ =
n−r ′ ′′ ′′
Fr+i X n−r −r −i Y n−r−r −i Z i . I(Pi , F ′ ∩ Z) then for non fundamental points {P1 , P2 ...Ps } on C ′ ∩ L, then
P
i=0
I(Pi , C ′ ∩ L) =
P P
I(Pi , Fr (X, Y ) ∩ L) = r, not only that each point Pi correspond to a tangent of C at P
and I(Pi , F ′ ∩ Z) is the multiplicity of the respective tangent. Which means if P doesn’t lie on C there are no
non-fundamental points in C ′ ∩ L. (c) is a direct consequence of this result since the only fundamental point
lying on C is P and finally by noting that C ′ is in good position and (C ′ )′ = C, (c) is proved. □
s
If C is a curve in excellent position, one has the following g ∗ (C ′ ) = g ∗ (C) − i=1 ri (ri − 1)/2 where ri are
P
multiplicities of non-fundamental points in C ′ ∩ L. If one could prove for a given curve F the existence of a
projective change of co-ordinates such that F T is in excellent position, Then by consequently performing the
quadratic transformations centered at the non-ordinary multiple points (After bringing the curve to an excellent
position) of the curve one either decreases the number of non-ordinary multiple points or decreases g ∗ . Since
g ∗ is non- negative, one gets a planar curve with only ordinary multiple points in N + g ∗ (C) steps where N
is the number of non-ordinary multiple points of F. Once this is achieved one gets a non-singular curve after
performing a blowup centered at ordinary multiple points. This could be easily seen as follows:
Theorem 8.3. (char(k)=0) Given a projective plane curve (irreducible) C = V (F ) and a point P on it such
that mP (C) = r, all but finitely many lines through P, intersect C in n − r distinct points.
Proof. Take P = [0 : 1 : 0]. Then consider lines of form Lλ = {(λ, t, 1)|t ∈ k} ∪ P (since the only other line is
z = 0). Now consider the polynomials of form Gλ (t) = F (λ, t, 1) which has degree n-r. Gλ has to be irreducible
since otherwise substituting λ = x and homogenising the coefficients of t in its factors, we get a factorisation
of F which is irreducible. Thus whenever the leading coefficient of Gλ is non-zero (which happens for all but
finitely many λ ∈ k) we have, the line Lλ to be intersecting C at n − r distinct points. □
This means one can easily find lines L, L′ and L′′ such that L · C = P1 + P2 + ...Pn , L′ · C = r · P + P2′ + ...Pn−r
and L′′ · C = r · P + P2′′ + ...Pn−r
′′
. Now we can take L to X; L′ to Y and L′′ to Z and we have C at the excellent
position with some co-ordinate change T with T ([0 : 0 : 1]) = P . Thus the existence of a non-singular model
of a curve is proved. A curve has a unique non-singular model upto isomorphism because isomorphism classes
and birational equivalence classes of non-singular curves coincide.
Theorem 8.4. Let C be an irreducible plane curve and f : X → C be the birational
P map from the non-singular
model X to C as constructed above. Then for a plane curve G, I(P, C ∩ G) = Q∈f −1 (P ) ordQ (G)
Proof. Choose an affine neighbourhood U of P such that g (image of G in Γ(U ) is a unit in the local rings
corresponding to all other points in in it. U ′ = f −1 (U ) be the affine open subset of X. Then I(P, C ∩ G) =
′ ′ ′
P k (Γ(U )/g). V = Γ(U ) ⊆ Γ(U ) = V , T (z) = gz is an injective, Thus dimk (Γ(U )/(g)) = dimk (Γ(U )/(g)) =
dim
Q∈f −1 (P ) ordQ (G) □

9. Rational Functions on non-singular curves

Suppose g is a rational function on a non-singular curve X, given a rational function g on X, one could assign
for each point P, ordP (g) since every local ring of the curve is a DVR. Hence one can
P assign for a rational function
on X an element in the free abelian group over the set {P ∈ X} such that g → ordP (g)P . Elements of this
P ∈X
10 NINTH SEMESTER THESIS

free abelian group are called divisors. Denote the divisor corresponding to the g by div(g). As one has already
seen, ordP (g) = 0; ordP (g) > 0; ordP (g) < 0 corresponds to points where g is non-zero; zero and where g has
a pole respectively thus allowing one to define the order of a pole or zero of a rational function. The following
theorem proves that not all divisors correspond to some rational function of X.
P
Theorem 9.1. If g is a rational function on X then ordP (g) = 0
P ∈X

Proof. One could also assign a divisor to a plane curve F (corresponding to some form in the homogenous
co-ordinate ring) on X since F∗ is a rational function on the affine patch of X containing P say X∗ and one
can define ordP (F ) to be ordX∗
P
P (F∗ ) . By Bezout’s theorem and 8.4 ordP (F ) = mn, where m = deg(G)
P ∈X
and n = deg(C) (C is the plane curve birationally equivalent to C). Define degree of a divisor to be the sum
of integers corresponding to the points in D say deg(D). A rational function g can be written in the form
f /h where f and h are elements in Γh (C) corresponds to plane curves F and H of the same degree. Then
deg(div(g)) = deg(div(F )) − deg(div(H)) = 0. □
Given a divisor D, one could ask what are the rational functions f in X such that div(f ) + D ≥ 0 where 0
refers to the divisor with all the coefficients 0 (D1 ≥ D2 if coefficients of all the points of D1 are greater than
that of D1 ). By virtue of properties of DVR, This space (denoted L(D)) is a vector space over k.
Theorem 9.2. L(D) is finite dimensional over k
Proof. One can easily prove this via induction on deg(D). First begin by noting that L(0) = k. Now assume
the statement to be true all D such that deg(D) < n where n ≥ 0. Consider the map ϕ : L(D + P ) → L(D)
by ϕ(f ) = tr+1 f (P ) where t is the uniforming parameter of OP (X) and r is the coefficient of P in D. This
is a well defined vector space morphism with ker(ϕ) = L(D). Thus dimk L(D + P ) ≤ dimk L(D) + 1. Thus
l(D) < deg(D) + 1 (where l(D) is dimk L(D)). □
Remark 9.3. If one considers LS (D) which is {f ∈ K| ordP (F ) ≥ −nP for P ∈ S} (D =
P
nP P ) for some
P ∈X
finite subset S of X then dimk LS (D + P )/LS (D) = 1. This is because we can always find a rational function z
for points P = Q0 , Q1 , Q2 ...Qs and integers m0 , m1 , m2 ..mr such that ordP (z) = m0 and ordQi (z) = mi (Once
we prove the fact for a plane curve C with only ordinary multiple points and birationally equivalent to X, we
are done. This is because we can choose a birational map for a finite set S of points in X such that the images of
each point in S are simple in C. So for points P1 , P2 , P3 ,...,Pn , choose lines Li passing through but
P not tangent
to Pi and also not passing through Pj ̸= Pi and L0 not passing through any Pi . z = Lm −mi
Q
i L satisfies
i

these conditions.). Set m0 = −nP − 1 and mi ≥ −nQ , thus ϕ mentioned in the previous theorem is surjective
and one has the result. The divisors D and D′ are said to be linearly equivalent if one has an f ∈ K such that
D = D′ + div(f ) (denoted as D ≡ D′ ). For such divisors the multiplication by f is an isomorphism of vector
spaces from L(D) to L(D′ ) hence l(D) = l(D′ ) and deg(D) = deg(D′ ).
It is evident that if D ≥ D′ , s(D) ≥ s(D′ ) where s(D) = deg(D) − l(D) + 1. In fact if one could prove that
this number is bounded for certain divisors on a given curve
P X, one could prove it for all divisors on the curve.
This could be seen as follows: Consider a divisor D = P ∈X mP P and an effective divisor Z such that there
exist some x ∈ K = k(X) such that Z = (x)0 , the divisors corresponding Q to zeroes ofmPx. Then for points in
T = {P ∈ X| ordP (x) ≤ 0 and mP > 0} one has a rational function f = (y − y(P )) where y = x−1 such
P ∈T
that D′ = D − div(f ) ≤ rZ for some large value of r. s(D) ≤ s(rZ) for large r which is bounded by the bound
for effective divisors. The following theorem will prove the boundedness of s for effective divisors.
Theorem 9.4. For an effective divisor Z = (x)0 of degree n, one has that l(rZ) ≥ rn − τ for all r and a
constant τ .
Proof. Let [K : k(x)] = m. Suppose {w1 , w2 , ..., wm } are basis of K over k(x) such that they are integral over
k[x−1 ]. Let S = {P ∈ X| ordP (x) > 0}, then for every P ∈ X either P ∈ S or ordP (wi ) ≥ 0 (Apply the
condition of order function of a DVR w.r.t ordP to the integral equations of wi over k[x−1 ]). This means there
is a t > 0 such that div(wi ) + tZ ≥ 0 for all i ≤ m. The set {wi x−j | 1 ≥ i ≥ m; 0 ≥ j ≥ r} are linearly
independent over k and lie in L((r + t)Z), thus n(r + t) + 1 ≥ l((r + t)Z) ≥ m(r + 1) where n = deg(Z). Taking
r → ∞ one also sees that n ≤ m. But also m ≤ n because of the following reason. If {v1 , v2 ...vn } are functions
in LS (0) such that their residues form a basis of LS (0)/Ls (−Z), they are linearly independent over k(x). As for
l(rZ) one has l(rZ) = l((r + t)Z) − dimk (L((r + t)Z)/L(rZ)) ≥ mr + (m − tn) = nr − (t − 1)n take τ = (t − 1)n
and the result is proved. □
One sees that for any divisor Z, There exists a constant g’ such that g ′ ≥ s(rZ) for all r. Let g the smallest
such constant, then one notes due to the property of s(D) that s(rZ) = g for sufficiently large r. Thus g is
also the bound {s(D)| D is a divisor in X}. This number corresponding to the curve X is called the genus
 NINTH SEMESTER THESIS 11

of X. The result about the boundedness of s(D) for all divisors is called the Riemann’s theorem. A curve
is rational if and only if it has genus 0. Consider a divisor D′ such that s(D′ ) = g, then for any divisor D
with deg(D) ≥ n + deg(D′ ) + g one sees that L(D − D′ ) has dimension greater than 0, which suggests (due to
Riemann’s theorem) the existence of f ∈ K, D + div(f ) ≥ D′ which implies g ≥ s(D) ≥ s(D′ ) ≥ g. Thus any
divisor D with a sufficiently large degree has s(D) = g.
9.1. The Genus Formula. Using the fact that g = deg(D) + 1 − l(D) for quite large divisors, one can find the
genus of a curve if one somehow manage to calculate l(D) accurately for some particular class divisors which
has divisors of degree larger than any given number. For this purpose consider a curve C with only ordinary
multiple points birationally equivalent to X and assume that Pit is in excellent position. Then Z = 0 intersects
it in P1 , P2 ...Pn n is the degree of the curve C. Let Em = m Qi − E where Qi are centered at Pi and E is the
same as the one mentioned in 11.3 and consider the following:
ψ ϕ
0 → Wm−n −
→ Vm −
→ L(Em ) → 0
Wm−n are the forms of degree m − n, Vm is the space of curves adjoint to C, ψ(H) = F H and ϕ(G) = G/Z m .
Given f ∈ L(Em ), if f = R/S then div(RZ m ) = div(S)+E, which means by 11.3 one has RZ m = AS +BF and
R/S = A/Z m in k(C). This means the above sequence is exact and thus l(Em ) + dimk (Wm−n ) − dimk (Vm ) = 0.
P mP (mP −1)
By 11.1 and 11.3, one gets that for large m, dimk (Vm ) = (m+1)(m+2)
2 − 2 . After substituting and
applying Riemann’s theorem one gets the genus of the curve equal to:
(n − 1)(n − 2) X mP (mP − 1)
g(X) = − = g ∗ (C)
2 2
10. Differentials on a Curve
R is a ring containing k and M is an R-module, then a map D : R → M such that D(xy) = xdy + ydx is
called a derivation on R into M. There exists a unique R-module Ωk (R) such that any derivation D on R to M
factors through Ωk (R) via an R-module homomorphism ϕD : Ωk (R) → M . This is so because one can define
a free module on symbols [x] corresponding to elements x of R and take its quotient with respect to module
generated by elements of form: [xy] − x[y] − y[x]; [λx] − λ[x]; [x + y] − [x] − [y] which will have the required
property and any other such R-module having this property is isomorphic as an R-module to this R-module.
Ωk (R) is called module of differentials of R over k If R is a domain with quotient field K, then any derivation
of R can be extended uniquely to K.
Proposition 10.1. If K is a function field of dimension 1 over k, Then Ωk (K) is a 1-dimensional vector space
over K.
Proof. Since any curve is birationally equivalent to a plane curve, one can assume R = k[X, Y ]/I of whom K is
a quotient field, where I = V (F ). Then since F (x, y) = 0 (x,y are residues of X and Y in R), FX (x, y)D(x) +
FY (x, y)D(y) = 0 for all derivations on R and hence FX (x, y)dx + FY (x, y)dy = 0, where dx and dy are elements
on Ωk (K) corresponding to x and y. It was already clear that Ωk (K) were generated by dx and dy but from
this relation, dy = udx where u = −FX (x, y)/FY (x, y). To prove that Ωk (K) is not zero consider the derivation
on R, D(G) = GX (x, y) − uGY (x, y), which has value 1 at x, since D̃ : K → k factors through Ωk (K) it can’t
be 0. □
Given ω ∈ Ω = Ωk (K) one can define the divisor of ω as follows: For P ∈ X, let t be a uniformizing parameter
of OP (X), then ordP (ω) = ordP (f ), to show that this is well defined consider another uniformizing parameter
of OP (X) say u and f dt = gdu. Let N be such that ordP (dx/dt) and ordP (dy/dt) are greater than or equal to
NP−1
−N , then for every g ∈ OP (X), ordP (dg/dt) ≥ −N . Let u = λi ti + tN g for some g ∈ OP (X), this means
i=1 P
du/dt ∈ OP (X) and similarly dt/du ∈ OP (X) which means ordP (f ) = ordP (g). Define div(ω) = ordP (ω)P .
The divisor which is divisor of a differential is called a canonical divisor.
Theorem 10.2. C is a plane curve of degree n ≥ 3, then div(G) − E is a canonical divisor, where G is a plane
curve of degree n − 3 and E is the same as mentioned in 11.3.
Proof. Assume C to be intersecting Z at {P1 , P2 , ..., Pn } and that no tangent at any multiple point passes
through [1 : 0 : 0] ∈ / C. Without loss of generality
P assume ω = dx, it will be sufficient to to show that
div(ω) = En−3 + div(FY /Z n−1 ) (where Em = P m Qi − E, Qi are points centered at Pi ) which is same as
div(dy) − div(FX ) = div(dx) − div(FY ) = −2 Pi − E
Case I: Q is centered at a point in Z ∩ C, then y −1 = Z/Y is a uniformizing parameter of mcOQ (X) and thus
ordQ (dy) = −2 since Fx is non-zero by Euler’s theorem (XFX + Y FY + ZFZ = nF ). ordQ matches on both
sides of the above equation.
Case II: If Q is centered at a point not in Z = 0, further two more case arise (AssumeP = [0 : 0 : 1] since the
differential dx won’t change on changing the origin ):
12 NINTH SEMESTER THESIS

subcase(i): If Y-axis is a tangent at P, it is not a multiple point and thus x is a uniformizing parameter of
OQ (X) and FY ̸= 0 thus again the order of Q matches at both sides of the equation and equal 0.
subcase(ii): If Y-axis is not a tangent at P, y is the u.p at OQ (X) and ordQ (dy) = 0 and ordQ (FX ) = mP (C)−1
since FX and F do not share any common tangents at P as P is at most an ordinary multiple point. (see 11.3
for details). □
From the application of Genus formula one gets that for any canonical divisor W one has deg(W ) = 2g − 2
and l(W ) ≥ g. Using canonical divisors one can calculate the exact value of l(D) thus bridging the gap in
Riemann’s theorem. The following lemma is the most important step towards the direction.
Lemma 10.3. l(D) > 0 and l(W − D − P ) ̸= l(W − D) then l(D + P ) = l(D).
Proof. Let C be such that Z intersects C (birationally equivalent to X and has only ordinary multiple points) at
n
P
{P1 , P2 , ..., Pn } (n is the degree of C). Em = m Pi − E and E is the same as defined before. WLOG assume
i=1
W = En−3 and D ≥ 0. Thus L(W − D) ⊂ L(En−3 ). Assume f (P ) to be simple in C. h ∈ L(W − D) − L(W −
D − P ) can be written as G/Z n−3 where div(G) = D + E + A and G is adjoint to C. One can choose a line L
passing through f(P) which intersects C at n − 1 other points such that div(LG) = D + P + E + A + B where
B is a divisor consisting of n − 1 points each of coefficient 1. One needs to show that for any f ∈ L(D + P ),
div(f )+L(D) = D′ ≥ 0. By residue theorem, there exists H of degree n−2 such that div(H) = D′ +P +E+A+B.
B consists of points centered at simple points in L∩C which means H intersects L at more than n−1 points, but
by Bezout’s theorem, L is then a component of H and thus H(f (P )) = 0, so div(H) = D′ + P + E + A + B ≥ P
which implies D′ ≥ 0 as none of E,A or B has P in it. □
One can now finally prove the Riemann-Roch Theorem which is that for any divisor D on X one has
l(D) = deg(D) + 1 − g + l(W − D) where W is a canonical divisor on X.
Case I: If l(W − D) = 0, one can do induction on l(D). If l(D) = 0 or l(D) = 1 this theorem can be proved
using Riemann’s theorem. If l(D) > 1 one can choose P such that l(D − P ) = l(D) − 1 (There are at most
a finite number of points P for which this doesn’t hold) , since l(D − P ) > 0 one can use 10.3 to show that
l(W − (D − P )) = 0. Now using induction hypothesis the theorem holds for l(D − P ) the theorem can be shown
to be true for D using the equation for D − P .
Case II: If l(W − D) > 0, one has got a maximal D (if any exists) for which the theorem is false (Since deg(D)
is bounded by deg(W) otherwise l(W − D) = 0 for which the theorem is already proved). Now one can apply
induction on l(D) again. The theorem can be proved for l(D) = 0 by applying Case I for W − D. If l(D) > 0
choose P such that l(W − D − P ) = l(W − D) − 1, then by 10.3, l(D) = l(D + P ) but by maximality of D, one
has that the theorem is true for D+P. Simplifying the equation, one gets the equation for D.

11. Appendix
11.1. Proof that ring of local rational functions is a localisation of the co-ordinate ring. First one
must start by observing that given a variety V and a point on it P, one can consider it to be an open subvariety
of a projective variety (say Pn ). Thus given a point on it any rational function on OP (V∗ ) can be extended
uniquely to whole of V minus a closed subset by homogenising (with respect to projective co-ordinates) the
polynomials in the numerator and denominator of the rational function, thus OP (V ) ∼ = OP (V∗ ) (V denotes the
intersection of V with the appropriate Ui ). As said earlier the stated fact can be easily proven for the affine case.
For the general case replace V by its closure in the projective space. Now one has k[x0 /xi , x1 /xi , ...xn /xi ] to be
localisation of R = k[x0 , x1 , ...xn ] at R − {xi , x2i ...}. Which would mean OP (V ) ∼
= OP (V∗ ) due to transitivity
of localisations.
11.2. An inequality regarding d-forms in k[x, y, z]. The d-forms in k[x,y,z] are a vector space over k the
basis being the monomials of degree d which are (d+1)(d+2)
2 in number but since the multiplication by an element
in k represent the same form, they form a projective space of dimension N = d(d+3) 2 . d − f orms passing through
2 N
a point in P form a hyperplane in P . If P = [0 : 0 : 1], then the plane curves of degree d with mP (C) ≥ r
correspond to forms in which the coefficients of monomials with sum of the X and Y powers is less than r is 0
are r(r + 1)/2 in number. Hence it is a linear subvariety of Pn of dimension d(d+3) 2 − r(r+1)
2 . By the dimension
formula proved before the dimension of the linear subvariety V (d; r1 P1 , ...rn Pn ) = { Curves F of degree d such
that mPi (F ) ≥ ri } is bounded below by d(d+3)
P ri (ri +1)
2 − 2 .
For any projective plane curve of degree n we have as a consequence of Bezout’s theorem the following: n(n−1) 2 −
P mP (mP −1)
2 ≥ 0.
n(n + 3) X mP (mP − 1) n(n − 1) X mP (mP − 1)
r= − ≥ − ≥0
2 2 2 2
. Choose simple points Q1 , Q2 , ..., Qr in F then we have a curve G passing through all multiple points of F
with the multiplicity mP − 1 and passing through Q1 , Q2 ..Qr with multiplicity 1. Using bezout’s formula for
 NINTH SEMESTER THESIS 13

mP (mP − 1) + r. Substituting r one gets g ∗ (C) = (n − 1)(n − 2)/2 −

P
their intersection, one gets n(n − 1) ≥
P mP (mP −1)
2 ≥ 0.

Remark 11.1. For d ≥ ( ri ) − 1, V (d; r1 P1 , ...rn Pn ) = d(d+3)

P P ri (ri +1)
2 − 2 .
11.3. Residue Theorem. Consider a plane curve C which has only ordinary multiple points with its non-
singular model X and f : X → C be the birational map from X to C obtained as in 8. Let C1 , C2 ...Cn be the
curves which are obtained one after the other as in the fashion mentioned in 8 and Cn be the plane curve with
only ordinary multiple points. For every ordinary multiple point P in C, there is an open affine neighbourhood
W which is isomorphic to an open affine set in Cn . Which means there will be an open affine set of X say
W’ such that it will be the blowup of W at P and the blowup map being f . Now consider an affine plane
curve C̃ such that X = 0 is not a tangent at Q which is an ordinary multiple point and its blowup C̃ ′ at
Q. Γ(C̃) ⊂ Γ(C̃ ′ ) via the injective map f → f ◦ ϕ (since the blowup map induces injective homomorphism
′
of function fields) where ϕ(x, z) = (x, xz). If G is a plane curve and g is its image in Γ(C̃) Then ordC̃ Qi (g) =
I(Qi , G(X, XZ)∩ C̃ ′ ) = I(Qi , X mP (G) ∩C ′ )+I(Qi , G′ ∩ C̃ ′ ) ≥ mP (G) (where Qi are points in ϕ−1 (Q). Equality
holds if G and C̃ doesn’t have any common tangents. From the previous discussion one could say that the same
holds for C and the point P. Which means if f −1 (P ) = {P1 , P2 , P3 ...Pr } and for s ≤ r if one has ordX Pi (g) ≥ s,
then mP (G) ≥ s since otherwise we would get a contradiction that mP (G) ≥ r ≥ sP and mP (G) < s. Thus for
the plane curve C with only ordinary multiple points, one has that div(G) ≥ E = Q∈X (mf (Q) − 1)Q if and
only if mP (G) ≥ mP (C) − 1. When this happens G is said to be adjoint to C.
Consider projective plane curves F,G and H. We can write H = AF + BG where A and B are forms, if and
only if at each point in F ∩ G, H∗ ∈ (F∗ , G∗ ). If the triplet (F,G,H) satisfy this condition at a point P ∈ F ∩
they are said to satify noether’s condition at P. Let C be a plane projective curve given by F. Then F,G,H
satisfy noether’s condition at P, if z = H ∗ /G∗ ∈ OP (F ). If P is ordinary multiple point with multiplicity r and
ordPi (z) ≥ r − 1 at all Pi then one can show using 8.1 that z ∈ OP (C). That is if ordPi (H) ≥ ordPi (G) + r − 1.
Consider effective divisors D and D′ such that D ≡ D′ and there exists a curve G adjoint to C such that
div(G) = D + E + A where A is an effective divisor, then div(GH) = D′ + div(H ′ ) + E + A ≥ div(H ′ ) + E (H
and H’ are forms such that div(H) + D = div(H ′ ) + D′ ). From the last paragraph of seection 8, {GH, H ′ , F }
satisfies noether’s conditions at all points of C so one can write GH = F ′ F + G′ H ′ and div(G′ ) = D′ + E + A.
The fact that such a G’ corresponding to D’ exists is called the Residue Theorem.
11.4. Coordinate change. Let T = (T1 , T2 , T3 , ..., Tm ) be a polynomial map from An → Am . If V is an
algebraic set in Am then one can define V T in An to be the variety corresponding to ideal in k[x1 , x2 , ..., xn ]
generated by polynomials f ◦ T where f is a polynomial in the ideal in k[y1 , y2 , ..., ym ] corresponding to V. If
m = n and Ti are linear such that T is a bijection, then T is said to be a co-ordinate change of An . One can
similarly define a projective change of co-ordinates where Ti are homogenous polynomials of degree 1. These are
some proofs and results regarding and using the change of co-ordinates which were used extensively throughout
the text for proving various important results.
a) P1 , P2 , and P3 (resp. Q1 , Q2 , and Q3 ) are non-collinear points in P2 , then there exist co-ordinate change T
such that T (Pi ) = Qi
b) Let L1 ,L2 and L3 (resp. M1 , M2 , and M3 ) be non-concurrent, then there is a projective change of co-ordinates
T such that T (Li ) = Mi
11.5. Proof of integral extension property of blowups: Consider the birational map ψ : A2 → A2 ,
ψ(x, z) = (x, xz). As proved before one can take The closure C ′ of ψ −1 (C) a curve C passing through origin
and ψ|C ′ : C ′ → C will be a birational map of curves. Now one can for the point P = (0, 0) in C consider an
ai0 X i−r (It is assumed that C = V (F = aij X i Y j ))
P P
affine neighbourhood given by W = Ch where H =
i≥r i+j≥r
assuming X is not tangent to C one can safely assume H(0, 0) ̸= 0 thus we also get an affine open set W ′ = Ch′
such that ψ(W ′ ) = W . One easily deduces that Γ(C ′ ) = Γ(C)(z), z = y/x where y and x is the image of Y
and X in Γ(C). Using the equation Of curve (F=0) one deduces that z satisfies an equation of degree r (The
degree of the lowest form of F) with the leading coefficient h over Γ(C), implying that it satisfies a polynomial
equation of degree r and leading coefficient one over Γ(Ch ) = Γ(C)[1/h]. Hence Γ(W ′ ) is integral over Γ(W )
and xr−1 Γ(W ′ ) ∈ Γ(W ).
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note series. Vol. 30 (3rd ed.). Addison-Wesley. ISBN 978-0-201-51010-2.

2) Hartshorne, Robin (1977). Algebraic Geometry. Berlin, New York: Springer-Verlag. ISBN 978-0-387-
90244-9