By Robert Friedman

A unique function of the e-book is its built-in method of algebraic floor concept and the learn of vector package conception on either curves and surfaces. whereas the 2 matters stay separate throughout the first few chapters, they develop into even more tightly interconnected because the e-book progresses. therefore vector bundles over curves are studied to appreciate governed surfaces, after which reappear within the facts of Bogomolov's inequality for strong bundles, that is itself utilized to review canonical embeddings of surfaces through Reider's process. equally, governed and elliptic surfaces are mentioned intimately, sooner than the geometry of vector bundles over such surfaces is analysed. a few of the effects on vector bundles seem for the 1st time in booklet shape, subsidized via many examples, either one of surfaces and vector bundles, and over a hundred routines forming an essential component of the textual content. aimed toward graduates with a radical first-year direction in algebraic geometry, in addition to extra complex scholars and researchers within the parts of algebraic geometry, gauge thought, or 4-manifold topology, a number of the effects on vector bundles can also be of curiosity to physicists learning string conception.

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**Extra info for Algebraic surfaces and holomorphic vector bundles**

**Example text**

1. i/ D X \ PXni be the corresponding affine chart. i/ /, i D 1; : : : ; n C 1. Proof. Suppose for example that i D n C 1. X / ! nC1/ /, and it is easy to see that it is an isomorphism of K-algebras. 6. nC1/ . 4). X1 ; : : : ; XnC1 / where G, H are homogeneous polynomials of the same degree in KŒX1 ; : : : ; XnC1 . X /. Rational maps between projective (or affine) varieties are defined by way of rational functions. If X P n is an irreducible algebraic set, then a rational map (or rational transformation) ' W X !

X1 ; : : : ; XnC1 / where G, H are homogeneous polynomials of the same degree in KŒX1 ; : : : ; XnC1 . X /. Rational maps between projective (or affine) varieties are defined by way of rational functions. If X P n is an irreducible algebraic set, then a rational map (or rational transformation) ' W X ! X /. fj /. A rational map (or rational transformation) ' W X ! X / and it is well defined on the set, which is an open dense subset of X (cf. X / is a non-zero element, then gf1 ; : : : ; gfmC1 define the same rational map.

Dom. // W then W X ! W is a rational map between the two algebraic sets X and W . The map W X ! dom. dom. // D W . We note that, given two rational maps W X ! W , W W ! Z between algebraic sets, one can then consider the rational map B W X ! dom. // \ dom. / ¤ ;. In particular, this is always the case if is dominant. 4. 5. Let W X ! X / dense in W , and W W ! Z be rational transformations between algebraic sets. If Im. / is dense in Z then also Im. B / is dense in Z. It follows that . B / D B .