orbifolds there is a non-zero algebraic basis Œª C 1 , . . . , Œª C n with rational coeÔ¨Écients of H 1 , 1 prim ( X, Q ) , that is, there are n ‚à∂ = h 1 , 1 prim ( X, Q ) algebraic curves C 1 , . . . , C n in X such that under the Poincar¬¥e duality the class in homology [ C i ] goes to Œª C i , [ C i ] ‚Ü¶ Œª C i . Recall that the Cox ring of P k + 2 is contained in the Cox ring of P 2 k + 1 Œ£ ,X without considering the grading. Considering the grading we have that if Œ± ‚àà Cl ( P k + 2 Œ£ ) then ( Œ±, 0 ) ‚àà Cl ( P 2 k + 1 Œ£ ,X ) . So the polynomials deÔ¨Åning C i ‚äÇ P k + 2 Œ£ can be interpreted in P 2 k + 1 X, Œ£ but with diÔ¨Äerent degree. Moreover, by Remark 4.1 each C i is contained in Y = { F = y 1 f 1 + ‚ãØ + y k f k = 0 } and\n\nfurthermore it has codimension k .\n\nClaim: { C i } ni = 1 is a basis of prim ( ) . It is enough to prove that Œª C i is diÔ¨Äerent from zero in H k,k prim ( Y, Q ) or equivalently that the cohomology classes { Œª C i } ni = 1 do not come from the ambient space. By contradiction, let us assume that there exists a j and C ‚äÇ P 2 k + 1 Œ£ ,X such that Œª C ‚àà H k,k ( P 2 k + 1 Œ£ ,X , Q ) with i ‚àó ( Œª C ) = Œª C j or in terms of homology there exists a ( k + 2 ) -dimensional algebraic subvariety V ‚äÇ P 2 k + 1 Œ£ ,X such that V ‚à© Y = C j so they are equal as a homology class of P 2 k + 1 Œ£ ,X ,i.e., [ V ‚à© Y ] = [ C j ] . It is easy to check that œÄ ( V ) ‚à© X = C j as a subvariety of P k + 2 Œ£ where œÄ ‚à∂ ( x, y ) ‚Ü¶ x . Hence [ œÄ ( V ) ‚à© X ] = [ C j ] which is equivalent to say that Œª C j comes from P k + 2 Œ£ which contradicts the choice of [ C j ] .\n\nRemark 5.2 . Into the proof of the previous theorem, the key fact was that on X the Hodge conjecture holds and we translate it to Y by contradiction. So, using an analogous argument we have:\n\nargument we have:\n\nProposition 5.3. Let Y = { F = y 1 f s +‚ãØ+ y s f s = 0 } ‚äÇ P 2 k + 1 Œ£ ,X be the quasi-smooth hypersurface associated to a quasi-smooth
Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems.
Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems. ->: orbifolds there is a non-zero algebraic basis Œª C 1 , . . . , Œª C n with rational coeÔ¨Écients of H 1 , 1 prim ( X, Q ) , that is, there are n ‚à∂ = h 1 , 1 prim ( X, Q ) algebraic curves C 1 , . . . , C n in X such that under the Poincar¬¥e duality the class in homology [ C i ] goes to Œª C i , [ C i ] ‚Ü¶ Œª C i . Recall that the Cox ring of P k + 2 is contained in the Cox ring of P 2 k + 1 Œ£ ,X without considering the grading. Considering the grading we have that if Œ± ‚àà Cl ( P k + 2 Œ£ ) then ( Œ±, 0 ) ‚àà Cl ( P 2 k + 1 Œ£ ,X ) . So the polynomials deÔ¨Åning C i ‚äÇ P k + 2 Œ£ can be interpreted in P 2 k + 1 X, Œ£ but with diÔ¨Äerent degree. Moreover, by Remark 4.1 each C i is contained in Y = { F = y 1 f 1 + ‚ãØ + y k f k = 0 } and\n\nfurthermore it has codimension k .\n\nClaim: { C i } ni = 1 is a basis of prim ( ) . It is enough to prove that Œª C i is diÔ¨Äerent from zero in H k,k prim ( Y, Q ) or equivalently that the cohomology classes { Œª C i } ni = 1 do not come from the ambient space. By contradiction, let us assume that there exists a j and C ‚äÇ P 2 k + 1 Œ£ ,X such that Œª C ‚àà H k,k ( P 2 k + 1 Œ£ ,X , Q ) with i ‚àó ( Œª C ) = Œª C j or in terms of homology there exists a ( k + 2 ) -dimensional algebraic subvariety V ‚äÇ P 2 k + 1 Œ£ ,X such that V ‚à© Y = C j so they are equal as a homology class of P 2 k + 1 Œ£ ,X ,i.e., [ V ‚à© Y ] = [ C j ] . It is easy to check that œÄ ( V ) ‚à© X = C j as a subvariety of P k + 2 Œ£ where œÄ ‚à∂ ( x, y ) ‚Ü¶ x . Hence [ œÄ ( V ) ‚à© X ] = [ C j ] which is equivalent to say that Œª C j comes from P k + 2 Œ£ which contradicts the choice of [ C j ] .\n\nRemark 5.2 . Into the proof of the previous theorem, the key fact was that on X the Hodge conjecture holds and we translate it to Y by contradiction. So, using an analogous argument we have:\n\nargument we have:\n\nProposition 5.3. Let Y = { F = y 1 f s +‚ãØ+ y s f s = 0 } ‚äÇ P 2 k + 1 Œ£ ,X be the quasi-smooth hypersurface associated to a quasi-smooth