--- title: "Algebraic geometry" - CiteKey: "Har08" - Type: book - Title: "Algebraic geometry," - Author: "Hartshorne, Robin;" - Publisher: "Springer," - Year: 2008 - Collections: "Basic AG; Hartshorne Study; Textbooks; Textbooks," - Collections: "Hartshorne Study; Textbooks," --- # Algebraic geometry ## Meta - **URI**: - **Local File**: [Hartshorne - 2008 - Algebraic geometry.pdf](file:///home/zack/Zotero/storage/JYT94TJB/Hartshorne%20-%202008%20-%20Algebraic%20geometry.pdf) - **Open in Zotero**: [Zotero](zotero://select/library/items/DHH5AE7A) ## Abstract ---- ## Extracted Annotations Annotations(6/1/2022, 12:47:15 PM) ![[Har08_9GVC3MFW.png]] [(Hartshorne, 2008, p. 10)](zotero://open-pdf/library/items/JYT94TJB?page=10&annotation=CZV3P3NI) ![[Har08_BUAYFJF6.png]] [(Hartshorne, 2008, p. 18)](zotero://open-pdf/library/items/JYT94TJB?page=18&annotation=68ZSUUVF) ![[Har08_WEZTIW9P.png]] [(Hartshorne, 2008, p. 77)](zotero://open-pdf/library/items/JYT94TJB?page=77&annotation=U2CF2L44) ![[Har08_Q9S4AKBT.png]] [(Hartshorne, 2008, p. 218)](zotero://open-pdf/library/items/JYT94TJB?page=218&annotation=M6B59CH7) ![[Har08_GVUWDZ8S.png]] [(Hartshorne, 2008, p. 310)](zotero://open-pdf/library/items/JYT94TJB?page=310&annotation=EQ3N8CSY) ![[Har08_39NFDDE4.png]] [(Hartshorne, 2008, p. 373)](zotero://open-pdf/library/items/JYT94TJB?page=373&annotation=QJZI2JWU) - we treat two special classes of surfaces, the ruled surfaces, and the nonsingular cubic surfaces in P°, [(Hartshorne, 2008, p. 373)](zotero://open-pdf/library/items/JYT94TJB?page=373&annotation=LJLEPKGK) - As applications we give the Hodge index theorem and the Nakai-Moishezon criterion for an ample divisor. [(Hartshorne, 2008, p. 373)](zotero://open-pdf/library/items/JYT94TJB?page=373&annotation=FR3E7UZY) - we prove the theorem of factorization of a birational morphism i [(Hartshorne, 2008, p. 373)](zotero://open-pdf/library/items/JYT94TJB?page=373&annotation=MFFEK3PT) - prove Castelnuovo9s criterion for contracting an exceptional curve of the first kind. [(Hartshorne, 2008, p. 373)](zotero://open-pdf/library/items/JYT94TJB?page=373&annotation=SLVZBX92) - Here the theory of curves gives a good handle on the ruled surfaces, because many properties of the surface arc closely related to the study of certain linear systems on the base curve. [(Hartshorne, 2008, p. 373)](zotero://open-pdf/library/items/JYT94TJB?page=373&annotation=2VHUGG96) - there is a close connection between ruled surfaces over a curve C and locally free sheaves of rank 2 on C, [(Hartshorne, 2008, p. 373)](zotero://open-pdf/library/items/JYT94TJB?page=373&annotation=FX58GR6W) - e study the nonsingular cubic surfaces in P3, and the famous 27 1ineS which lie on those surfaces. By representing the surface as a P° with 6 points blown up, the study of linear systems on the cubic surface is reduced [(Hartshorne, 2008, p. 373)](zotero://open-pdf/library/items/JYT94TJB?page=373&annotation=22DQ3ALH) - to the study of certain linear system of plane curves with assigned base points. [(Hartshorne, 2008, p. 374)](zotero://open-pdf/library/items/JYT94TJB?page=374&annotation=BVAVYFI6) - he Riemann4 Roch theorem for surfaces gives a connection between the dimension of a complete linear system |D|, which is essentially a cohomological invariant, and certain intersection numbers on the surface. [(Hartshorne, 2008, p. 374)](zotero://open-pdf/library/items/JYT94TJB?page=374&annotation=PAPL2VJI) - surface will mean a nonsingular projective surface over an algebraically closed field k. [(Hartshorne, 2008, p. 374)](zotero://open-pdf/library/items/JYT94TJB?page=374&annotation=7ID8QMA7) - curve on a surface will mean any effective divisor on the surface. In particular, it may be singular, reducible or even have multiple components. A point will mean a closed point, unless otherwise specified. [(Hartshorne, 2008, p. 374)](zotero://open-pdf/library/items/JYT94TJB?page=374&annotation=9B2CG4PQ) - If C and D are curves on X, and if Pe C n D is a point of intersection of C and D, we say that C and D meet transversally at P if the local equations f,g of C,D at P generate the maximal ideal mp of Op x. [(Hartshorne, 2008, p. 374)](zotero://open-pdf/library/items/JYT94TJB?page=374&annotation=J6YLGHD7) - PrOOF. We embed X in a projective space P" using the very ample divisor D. [(Hartshorne, 2008, p. 375)](zotero://open-pdf/library/items/JYT94TJB?page=375&annotation=GXBXNAS3) - Then we apply Bertini9s theorem [(Hartshorne, 2008, p. 375)](zotero://open-pdf/library/items/JYT94TJB?page=375&annotation=HCCIK96J) ![[Har08_ARELWIKT.png]] [(Hartshorne, 2008, p. 375)](zotero://open-pdf/library/items/JYT94TJB?page=375&annotation=VCHBS8VY) - Here, of course, .#(D) is the invertible sheaf on X corresponding to D (I, §7), and deg. denotes the degree of the invertible sheaf ¥ (D) ® (¢ on C (IV,§1). [(Hartshorne, 2008, p. 375)](zotero://open-pdf/library/items/JYT94TJB?page=375&annotation=4R5X598K) - #(4 D) is the ideal sheaf of D on X. [(Hartshorne, 2008, p. 375)](zotero://open-pdf/library/items/JYT94TJB?page=375&annotation=7MSAKLL5) - f C and D are curves with no common irreducible component, and if P ¬ C n D, then we define the intersection multiplicity (C.D)p of C and D at P to be the length of (. x/( f.¢), where f,g are local equations of C.D at P (I, Ex 5.4). Here length is the same as the dimension of a k-vector space. [(Hartshorne, 2008, p. 377)](zotero://open-pdf/library/items/JYT94TJB?page=377&annotation=IM2Z5FGF) - Intersection number as a sum of local intersection multiplicities ![[Har08_CWAWP99G.png]] [(Hartshorne, 2008, p. 377)](zotero://open-pdf/library/items/JYT94TJB?page=377&annotation=TSD8RJ7E) - If D is any divisor on the surface X, we can define the selfintersection number D.D, usually denoted by D*. Evenif Cisa nonsingular curve on X, the self-intersection C? cannot be calculated by the direct metho [(Hartshorne, 2008, p. 377)](zotero://open-pdf/library/items/JYT94TJB?page=377&annotation=K8KP28I5) - How to compute self-intersection numbers as the degree of the normal sheaf $\mcn_{C, X}$. ![[Har08_9SE4BQEB.png]] [(Hartshorne, 2008, p. 378)](zotero://open-pdf/library/items/JYT94TJB?page=378&annotation=FBDYAHI4) - e must use linear equivalence. [(Hartshorne, 2008, p. 378)](zotero://open-pdf/library/items/JYT94TJB?page=378&annotation=5WMJSCFF) - Intersection numbers on P^2[ ](zotero://open-pdf/library/items/?page=undefined&annotation=) - Intersection numbers of quadric surfaces in P^3[ ](zotero://open-pdf/library/items/?page=undefined&annotation=) - Self-intersection of the canonical[ ](zotero://open-pdf/library/items/?page=undefined&annotation=) - Adjunction formula ![[Har08_D83BCMI9.png]] [(Hartshorne, 2008, p. 378)](zotero://open-pdf/library/items/JYT94TJB?page=378&annotation=5345A5RT) ![[Har08_8EVGXNCJ.png]] [(Hartshorne, 2008, p. 441)](zotero://open-pdf/library/items/JYT94TJB?page=441&annotation=KVHDKW37) - 1 Intersection Theory [(Hartshorne, 2008, p. 442)](zotero://open-pdf/library/items/JYT94TJB?page=442&annotation=QJPA7GWQ) ![[Har08_PMAECSS6.png]] [(Hartshorne, 2008, p. 443)](zotero://open-pdf/library/items/JYT94TJB?page=443&annotation=3KDBCVFS) - It is called the Chow ring of X. [(Hartshorne, 2008, p. 443)](zotero://open-pdf/library/items/JYT94TJB?page=443&annotation=NV89AJLU)- The Chow ring. - Properties of the Chow Ring [(Hartshorne, 2008, p. 445)](zotero://open-pdf/library/items/JYT94TJB?page=445&annotation=TX527D65) - 3 Chern Classes [(Hartshorne, 2008, p. 446)](zotero://open-pdf/library/items/JYT94TJB?page=446&annotation=Q3M277AA) - The Riemann-Roch Theorem [(Hartshorne, 2008, p. 448)](zotero://open-pdf/library/items/JYT94TJB?page=448&annotation=U42NKI7L) - Hirzebruch-Riemann-Roch theorem. ![[Har08_T6IA8N35.png]] [(Hartshorne, 2008, p. 449)](zotero://open-pdf/library/items/JYT94TJB?page=449&annotation=XBUC6UKH) ![[Har08_B9A9AFYY.png]] [(Hartshorne, 2008, p. 455)](zotero://open-pdf/library/items/JYT94TJB?page=455&annotation=8TBVKBEX) ![[Har08_VTH2YLZC.png]] [(Hartshorne, 2008, p. 466)](zotero://open-pdf/library/items/JYT94TJB?page=466&annotation=WWD6DRRR)