I 'm trying to solve this exercise of Fulton Algebraic Curves:Let D = ∑ n P P be an effective divisor, S = { P ∈ X : n P &gt; 0 }, U = X ∖ S. Show that L ( r D ) ⊂ Γ ( U , O X ). Where L ( E ) = { f ∈ K : ord P ( f ) &gt; − n P , ∀ P ∈ X } (The linear vectorial space of the rational functions with pole multiplicity worst at n P ).But I have no idea

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I &#039;m trying to solve this exercise of Fulton Algebraic Curves:Let   D  =  ∑      n    P    P be an effective divisor,   S  =  {  P  ∈  X  :      n    P    &amp;gt;  0  },   U  =  X  ∖  S. Show that   L  (  r  D  )  ⊂  Γ  (  U  ,            O        X    ). Where   L  (  E  )  =  {  f  ∈  K  :      ord    P    (  f  )  &amp;gt;  −      n    P    ,  ∀  P  ∈  X  } (The linear vectorial space of the rational functions with pole multiplicity worst at       n    P  ).But I have no idea

Kaiden Porter · Accepted Answer

Here&#039;s a hint:By definition, the   Γ  (  U  ,            O        X    ) are the regular functions on   U - in other words, the functions on   X that have no poles on   U.And here&#039;s a stronger hint, hidden for you (mouse-over if you really want to see):Also,   L  (  r  D  ) consist of the functions   f such that   d  i  v  (  f  )  +  r  D  ≥  0. This happens only if   f has at most poles of order   r      n    P   at   P and nowhere else. In particular, the elements of   L  (  r  D  ) are are regular functions on   U.

I 'm trying to solve this exercise of Fulton Algebraic Curves: Let D = ∑

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