By Oscar Zariski

The most objective of this e-book is to provide a totally algebraic method of the Enriques¿ category of tender projective surfaces outlined over an algebraically closed box of arbitrary attribute. This algebraic procedure is without doubt one of the novelties of this ebook one of the different smooth textbooks dedicated to this topic. chapters on floor singularities also are integrated. The booklet may be priceless as a textbook for a graduate path on surfaces, for researchers or graduate scholars in algebraic geometry, in addition to these mathematicians operating in algebraic geometry or similar fields"

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**Example text**

I=1 Example 6. Horrocks and Mumford [68] have shown that in P4 there is a 2-dimensional complex torus Y = C2 /Γ 4 2 (Γ = i=1 Zωi , ω1 , . . , ω4 ∈ C linearly independent over R). The tangent bundle to Y is trivial. From the exact sequence 0 → TY → TP4 |Y → NY /P4 → 0 it follows that det NY /P4 = OY (5). Thus to Y belongs a 2-bundle E over P4 with c1 (E) = 5, c2 (E) = 10. Hence Y is of degree 10. ) holomorphic 2-bundle over P4 which is known. In Chapter II, §3 we construct the bundle E without using the existence of non-singular abelian surfaces in P4 .

To each pair (ai , bi ) we choose polynomials pi ∈ H 0 (P3 , OP3 (ai )), qi ∈ H 0 (P3 , OP3 (bi )). Let Yi be the intersection of the hypersurfaces deﬁned by pi and qi . By choosing the polynomials pi and qi appropriately one can achieve that the intersections Yi are smooth and pairwise disjoint curves. Let Y be their union. The Koszul complex for Yi 0 → OP3 (−(ai + bi )) → OP3 (−ai ) ⊕ OP3 (−bi ) → JYi → 0 shows that the determinant bundle det NYi /P3 is isomorphic to OYi (ai + bi ) = OYi (p) for all i = 1, .

We now calculate Ext 1OPn (JY , OPn (−k)): from the Ext-sequence associated to 0 → JY → OPn → OY → 0 we get ∼ Ext 1OPn (JY , OPn (−k)) − → Ext 2OPn (OY , OPn (−k)). 50 1. HOLOMORPHIC VECTOR BUNDLES AND THE GEOMETRY OF Pn Since Y is a codimension 2 locally complete intersection, we have (Altman and Kleiman, [1], p. 12–14, Griﬃths and Harris, [49], p. 690–692) the local fundamental isomorphism (LFI) ∼ Ext 2OPn (OY , OPn (−k)) − → Hom OY (det JY /JY2 , OY (−k)), where OY (−k) = OPn (−k) ⊗ OY . However, by assumption det JY /JY2 = OY (−k), so Hom OY (det JY /JY2 , OY (−k)) OY .