# Download e-book for iPad: Contributions to Algebraic Geometry: Impanga Lecture Notes by Piotr Pragacz

By Piotr Pragacz

The articles during this quantity are the result of the Impanga convention on Algebraic Geometry in 2010 on the Banach heart in Będlewo. the next spectrum of themes is covered:

K3 surfaces and Enriques surfaces;

Prym types and their moduli;

invariants of singularities in birational geometry;

differential varieties on singular spaces;

minimum version Program;

linear systems;

toric varieties;

Seshadri and packing constants;

equivariant cohomology;

Thom polynomials;

mathematics questions.

The major objective of the amount is to offer accomplished introductions to the above subject matters via texts ranging from an hassle-free point and finishing with the dialogue of present learn. the 1st 4 themes are represented by means of the notes from the minicourses held through the convention. within the articles the reader will locate classical effects and techniques, in addition to glossy ones. The ebook is addressed to researchers and graduate scholars in algebraic geometry, singularity concept and algebraic topology. lots of the fabric uncovered within the quantity has now not but seemed in ebook shape.

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**Extra resources for Contributions to Algebraic Geometry: Impanga Lecture Notes**

**Example text**

2 Toric downgrades. The toric case, however, can also provide us with more interesting examples. ı; Nz / and fix a subtorus action T ,! Tz . ı; Nz / as a T -variety? Assuming that the embedding T ,! Tz is induced from a surjection of the corresponding character z ! groups p W M ! M , we denote the kernel by MY and obtain two mutually dual exact sequences (1) and (2). _ . On the dual Setting WD NQ \ ı, the map p gives us a surjection ı _ ! z side, denote the surjection N ! NY by q. ı/, we denote by † the coarsest fan refining the images of all faces of ı under the map q.

Süß, and R. Vollmert Proof. 8]. Remark 6. A special case of the above is when X and X 0 are toric varieties. Here, the 0 proposition simplifies to TV. / Š TV. / TV. 0 /. 1 Maps between toric varieties. We will now see that all constructions from the previous section are functorial. 1). A Z-linear map F W N 0 ! N satisfying FQ . F / W TV. 0 ; N 0 / ! TV. ; N / of affine toric varieties via F _ . _ \ M / Â . 0 /_ \ M 0 . For example, if E Â _ \ M is a Hilbert basis, then the embedding TV. / ,! 1) is induced by the map E W N !

N k/-dimensional variety Y which is a sort of quotient Y D X=T . Now, X can be described by presenting a “polyhedral” divisor D on Y with coefficients being not numbers but instead convex polyhedra in the vector space NQ WD N ˝Z Q where N is the lattice of one-parameter subgroups of T . 3 What this paper is about. The idea of the present paper is to give an introduction to this subject and to serve as a survey for the many recent papers on T -varieties. Moreover, since the notion of polyhedral divisors and the theory of T -varieties closely follows the concept of toric varieties, we will treat both cases in parallel.