We will give a treatment of the adjunction formula which unifies these From the Poincar´eresidue formula, the 3 holomorphic differentials ω1, ω2, ω3 on C are of the form
This admits an interpretation as a Numerical Adjunction Formula for singular curves on the weighted projective plane
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If Xis proper nonsingular and dimension n, then for 0 ≤i≤n, Hi(X,L) ⊗Hn−(X,K⊗L∨) →Hn(X,K) ∼C is a perfect pairing
Adjunction formula for singular hypersurface in
Suppose Y Y is a singular hypersurface in Pn P n whose singular locus of codimension ≥ 2 ≥ 2, then the
Here is a heuristic argument for
This follows from the adjunction formula, standard results on curves of arithmetic genus p a = 0 and straight forward computations with intersection numbers, see [Liu
3) Xc = Xx 0 <9o(C)
The familiar Riemann-Roch formula for a non-singular projective algebraic curve (equivalently in the com-plex case, a Riemann surface) equates algebraic/analytic
This admits an interpretation as a numerical adjunction formula for
This admits an interpretation as a Numerical Adjunction Formula for singular curves on the weighted projective plane
One interesting feature of the adjunction formula is that it suggests that instead of working with canonical divisors we ought to work with canonical divisors plus other divisors: Definition 2
as an image of a smooth projective curves
If the curve is non-singular the geometric genus and the arithmetic genus are equal, but if Roch for curves holds even for singular curves
I am using the word reduce because I am mainly The adjunction formula tells us that! X(D)j D ˘=! D: The above can be rephrased as (K X+ D)j D= K D in the language of divisors
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One way to do this is to form the projective closure X o= f[x;y;z] 2P2 jF(x;y;z) = 0g where F(x;y;z) = y2zd 2 Y (x a iz) = 0 is the homogenization of y2 f(x)
For x E C singular, y-l (x) consists of finitely many points corresponding to the different branches of C through x
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Writeπ = π1 ··· πs, where πi: Xi → Xi−1 is the blowing-up at a pointPi ∈ Xi−1 andX0 = P2,Xs = X
This admits an in-terpretation as a Numerical Adjunction Formula for singular curves on the weighted projective plane
Let \((X,L)\) be a smooth polarized variety of dimension \(n\) and consider an effective, irreducible divisor \(A\in |L|\)
adjunction formula ensures that the genus of a smooth symplectic surface in CP2 is determined by its degree (homology class)
A question on $\mathbb{P}^1$-fibration
Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site The canonical sheaf is S2
Unfortunately, this does not help with constructing genus 2 curves since we the different stated in the Adjunction Conjecture to the case codim(W,X) = 1
quotient singularities