Calculating of genus of a curve.Let C be a curve over F q in projective plane. So C can be done as zeroes of some gomogeneous polynomial ∈ F q [ x , y , z ] with degree n. Whether is there algorithm which is polynomial time in n that calculate arithmetic genus of C?

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Calculating of genus of a curve.Let C be a curve over             F        q   in projective plane. So C can be done as zeroes of some gomogeneous polynomial   ∈            F        q    [  x  ,  y  ,  z  ] with degree n. Whether is there algorithm which is polynomial time in n that calculate arithmetic genus of C?

Zoey Benitez · Accepted Answer

Step 1Since you have clarified your question, I can now answer.The arithmetic genus depends only on the degree of C and the dimension of the projective space it is embedded in.In fact, the arithmetic genus is defined as  1  −      P    C    (  0  )  ,where       P    C    (  t  ) is the Hilbert polynomial of C.But the Hilbert polynomial of a curve og degree d in             P        2   can be computed by the exact sequence  0  →            O                                P                2              (  −  d  )  →            O                                P                2              →            O        C    →  0.By additivity of exact sequences, we have      P    C    (  t  )  =                    (                            t        +        2            2                      )              −                    (                            t        −        d        +        2            2                      )              =  t  d  −      1    2        d    2    +      3    2    d  .Step 2Thus, we see that the arithmetic genus of any degree d curve (singular, reducible...) is given by  1  +      1    2        d    2    −      3    2    d  =            (      d      −      2      )      (      d      −      1      )        2    .In particular, a curve of degree 3 have arithmetic genus 1.

Calculating of genus of a curve. Let C be a curve over F_q in projective plane.

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