2   Some Remarks on Plane Curves  Work over K   K     Definition 2 1   Rational plane affine curve    A plane affine curve C     f   x   y     0       2 is rational  if it has a rational parametrisation  i e        t         t     K   t   such that    i    1     2  t         t         t     is injective on   1     finite set      ii  f       t         t       0   Example 2 2   a  Any  non singular  plane conic is rational   Substitute y   t   x   1    We get x 2   t 2   x   1   2   1  hence   x   1       x   1     t 2   x   1       0  hence x     1 or x   1   t 2 1   t 2   Therefore this has a rational parametrisation        x   y       1   t 2 1   t 2   2 t 1   t 2          b  Any singular plane cubic is rational  rational parametrisation   x   y       t 2   t 3     rational parametrisation    x   y       e x e r c i s e   e x e r c i s e      c  Corollary 1 6 shows that elliptic curves are not  rational   Remark 2 3   The genus g   C         0 is an invariant of a smooth projective curve C   If K     then g   C     genus of Riemann surface   A smooth plane curve C     2 of degree d has genus g   C       d   1     d   2   2   Proposition 2 4  Assuming that   K   K    C a smooth projective curve  Then    i  C is rational  see Definition 2 1   if and only if g   C     0    ii  C is an elliptic curve  see Definition 1 5   if and only if g   C     1   Proof   i  Omitted    ii     Check C a smooth plane curve  exercise   Then use Remark 2 3     See later      Order of vanishing  C algebraic curve  function field K   C    P   C smooth point   We write ord p   f   for the order of vanishing of f   K   C     at P  negative of f has a pole    Fact   ord p   K   C         is a discrete valuation  i e  ord p   f 1 f 2     ord p   f 1     ord p   f 2   and ord p   f 1   f 2     min   ord p   f 1     ord p   f 2       Definition   Uniformiser   t   K   C     is a uniformiser at P  if ord p   t     1   Example 2 5  C     g   0       2  g   K   x   y   irreducible   K   C     Frac K   x   y     g        g   g 0   g 1   x   y     g 2   x   y           where g i are homogeneous of degree i   Suppose P     0   0     C is a smooth point  i e  g 0   0  g 1   x   y       x     y with       not  both zero   Fact     x     y   K   C   is a uniformiser at P if and only if             0   Example 2 6       y 2   x   x   1     x             2     where     0   1  Projective closure  x   X Z  y   Y Z         Y 2 Z   X   X   Z     X     Z         2       Let P     0   1   0     Aim   Compute ord p   x   and ord p   y     Put w   Z Y  t   X Y  So     w   t   t   w     t     w           Now Pis the point   t   w       0   0    This is a smooth point with      ord p   t     ord p   t   w     ord p   t     w     1            implies ord p   w     3  Therefore ord p   x     ord p   t   w     1   3     2 and ord p   y     ord p   1   w       3   Riemann Roch Spaces  Let C be a smooth projective curve   Definition   Divisor   A divisor  is a formal sum of points on C  say     D     P   C n p P     where n p     and n p   0  for all but finitely many P   C   We write deg D     n P   We say D is effective  written D   0  if n P   0 for all P   If f   K   C      then  div   f       P   C ord p   f   P   The Riemann Roch space of D   Div   C   is      L   D       f   K   C       div   f     D   0       0         i e  the K vector space of rational functions on C with  poles no worse than specified by D    We quote  Riemann Roch for genus 1      dim L   D       deg D if deg D   0 0 or 1 if deg D   0 0 if deg D   0     For example  in Example 2 6      L   2 P       1   x   L   3 P       1   x   y       Proposition 2 7  Assuming that   K   K    char K   2  C     2 a smooth plane cubic  P   C a point of inflection  Then  we may change coordinates such that      C   Y 2 Z   X   X   Z     X     Z       for some     0   1 and P     0   1   0     Proof  We change coordinates such that P     0   1   0    T p   C       Z   0    and C     F   X   Y   Z     0       2   P   C part of inflection implies F   t   1   0     const t 3  i e  F  has no terms X 2 Y  X Y 2 or Y 3   Therefore      F     Y 2 Z   X Y Z   Y Z 2   X 3   X 2 Z   X Z 2   Z 3         The Y 2 Z coefficient must be   0 otherwise P   C is singular  and the coefficient of X 3 is   0 otherwize Z   F   We are free to rescale X  Y  Z and F  Then without loss of generality C is defined by      Y 2 Z   a 1 X Y Z   a 3 Y Z 2   X 3   a 2 X z Z   a 4 X Z 2   a 6 Z 3       Weierstrass form   Substituting Y   Y   1 2 a 1 X   1 2 a 3 Z  we may suppose a 1   a 3   0  Now      Y 2 Z   Z 3 f   X Z       for some monic cubic polynomial f   C is smooth  so f has distinct roots  Without loss of generality say 0   1      Then C is given by      Y 2 Z   X   X   Z     X     Z           Remark   It may be shown that the points of inflection on a smooth plane curve      C     F   X 1   X 2   X 3     0       2     are given by      F   det     2 F   X i   X j     Hessian   0       2 1   The degree of a morphism  Let     C 1   C 2 be a non constant morphism of smooth projective curves   Then       K   C 2     K   C 1    f   f       Definition   Degree of a morphism   deg     K   C 1         K   C 2       Definition   Separable morphism     is separable if K   C 1         K   C 2   is a separable field extension   Definition   Ramification index   Suppose P   C 1  Q   C 2      P   Q  Let t   K   C 2   be a uniformiser at Q   The ramification index of   at P is     e     P     ord P       t        always   1  independent of choice of t    Theorem 2 8  Assuming that       C 1   C 2 a non constant morphism of smooth projective curves  Then        P       1 Q e     P     deg     Q   C 2       Moreover  if   is separable then e     P     1 for all but finitely many P   C 1  In particular    i    is surjective  on K   points    ii        1   Q     deg     iii  If   is separable then equality holds in  ii  for all but finitely many Q   C 2   Remark 2 9   Let C be an algebraic curve  A rational map is given C       n  P     f 0   P     f 1   P         f n   P     where f 0   f 1       f n   K   C   are not all zero    Important Fact   If C is smooth then   is a morphism