paper goes the other way around, we translate some results for quasi-smooth intersection subvarieties to\n\nAcknowledgement. I thank Prof. Ugo Bruzzo and Tiago Fonseca for useful discus- sions. I also acknowledge support from FAPESP postdoctoral grant No. 2019/23499-7.\n\nLet M be a free abelian group of rank d , let N = Hom ( M, Z ) , and N R = N ‚äó Z R .\n\nif there exist k linearly independent primitive elements e\n\n, . . . , e k ‚àà N such that œÉ = { ¬µ\n\ne\n\n+ ‚ãØ + ¬µ k e k } . ‚Ä¢ The generators e i are integral if for every i and any nonnegative rational number ¬µ the product ¬µe i is in N only if ¬µ is an integer. ‚Ä¢ Given two rational simplicial cones œÉ , œÉ ‚Ä≤ one says that œÉ ‚Ä≤ is a face of œÉ ( œÉ ‚Ä≤ < œÉ ) if the set of integral generators of œÉ ‚Ä≤ is a subset of the set of integral generators of œÉ . ‚Ä¢ A Ô¨Ånite set Œ£ = { œÉ\n\n, . . . , œÉ t } of rational simplicial cones is called a rational simplicial complete d -dimensional fan if:\n\nall faces of cones in Œ£ are in Œ£ ;\n\nif œÉ, œÉ ‚Ä≤ ‚àà Œ£ then œÉ ‚à© œÉ ‚Ä≤ < œÉ and œÉ ‚à© œÉ ‚Ä≤ < œÉ ‚Ä≤ ;\n\nN R = œÉ\n\n‚à™ ‚ãÖ ‚ãÖ ‚ãÖ ‚à™ œÉ t .\n\nA rational simplicial complete d -dimensional fan Œ£ deÔ¨Ånes a d -dimensional toric variety P d Œ£ having only orbifold singularities which we assume to be projective. Moreover, T ‚à∂ = N ‚äó Z C ‚àó ‚âÉ ( C ‚àó ) d is the torus action on P d Œ£ . We denote by Œ£ ( i ) the i -dimensional cones\n\nFor a cone œÉ ‚àà Œ£, ÀÜ œÉ is the set of 1-dimensional cone in Œ£ that are not contained in œÉ\n\nand x ÀÜ œÉ ‚à∂ = ‚àè œÅ ‚àà ÀÜ œÉ x œÅ is the associated monomial in S .\n\nDeÔ¨Ånition 2.2. The irrelevant ideal of P d Œ£ is the monomial ideal B Œ£ ‚à∂ =< x ÀÜ œÉ ‚à£ œÉ ‚àà Œ£ > and the zero locus Z ( Œ£ ) ‚à∂ = V ( B Œ£ ) in the aÔ¨Éne space A d ‚à∂ = Spec ( S ) is the irrelevant locus.\n\nProposition 2.3 (Theorem 5.1.11 [5]) . The toric variety P d Œ£ is a categorical quotient A d ‚àñ Z ( Œ£ ) by the group Hom ( Cl ( Œ£ ) , C ‚àó ) and the group
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. ->: paper goes the other way around, we translate some results for quasi-smooth intersection subvarieties to\n\nAcknowledgement. I thank Prof. Ugo Bruzzo and Tiago Fonseca for useful discus- sions. I also acknowledge support from FAPESP postdoctoral grant No. 2019/23499-7.\n\nLet M be a free abelian group of rank d , let N = Hom ( M, Z ) , and N R = N ‚äó Z R .\n\nif there exist k linearly independent primitive elements e\n\n, . . . , e k ‚àà N such that œÉ = { ¬µ\n\ne\n\n+ ‚ãØ + ¬µ k e k } . ‚Ä¢ The generators e i are integral if for every i and any nonnegative rational number ¬µ the product ¬µe i is in N only if ¬µ is an integer. ‚Ä¢ Given two rational simplicial cones œÉ , œÉ ‚Ä≤ one says that œÉ ‚Ä≤ is a face of œÉ ( œÉ ‚Ä≤ < œÉ ) if the set of integral generators of œÉ ‚Ä≤ is a subset of the set of integral generators of œÉ . ‚Ä¢ A Ô¨Ånite set Œ£ = { œÉ\n\n, . . . , œÉ t } of rational simplicial cones is called a rational simplicial complete d -dimensional fan if:\n\nall faces of cones in Œ£ are in Œ£ ;\n\nif œÉ, œÉ ‚Ä≤ ‚àà Œ£ then œÉ ‚à© œÉ ‚Ä≤ < œÉ and œÉ ‚à© œÉ ‚Ä≤ < œÉ ‚Ä≤ ;\n\nN R = œÉ\n\n‚à™ ‚ãÖ ‚ãÖ ‚ãÖ ‚à™ œÉ t .\n\nA rational simplicial complete d -dimensional fan Œ£ deÔ¨Ånes a d -dimensional toric variety P d Œ£ having only orbifold singularities which we assume to be projective. Moreover, T ‚à∂ = N ‚äó Z C ‚àó ‚âÉ ( C ‚àó ) d is the torus action on P d Œ£ . We denote by Œ£ ( i ) the i -dimensional cones\n\nFor a cone œÉ ‚àà Œ£, ÀÜ œÉ is the set of 1-dimensional cone in Œ£ that are not contained in œÉ\n\nand x ÀÜ œÉ ‚à∂ = ‚àè œÅ ‚àà ÀÜ œÉ x œÅ is the associated monomial in S .\n\nDeÔ¨Ånition 2.2. The irrelevant ideal of P d Œ£ is the monomial ideal B Œ£ ‚à∂ =< x ÀÜ œÉ ‚à£ œÉ ‚àà Œ£ > and the zero locus Z ( Œ£ ) ‚à∂ = V ( B Œ£ ) in the aÔ¨Éne space A d ‚à∂ = Spec ( S ) is the irrelevant locus.\n\nProposition 2.3 (Theorem 5.1.11 [5]) . The toric variety P d Œ£ is a categorical quotient A d ‚àñ Z ( Œ£ ) by the group Hom ( Cl ( Œ£ ) , C ‚àó ) and the group