Projective geometry and homological algebra

@inproceedings{Eisenbud2002ProjectiveGA,
  title={Projective geometry and homological algebra},
  author={David Eisenbud},
  year={2002}
}
We provide an introduction to many of the homological commands in Macaulay 2 (modules, free resolutions, Ext and Tor. ..) by means of examples showing how to use homological tools to study projective varieties. 
Computing Amoebas
TLDR
This work gives explicit characterizations for the amoebas of classes of linear and nonlinear varieties and presents homotopy-based techniques to compute the boundary of two-dimensional amoEBas.
Enumerative Algebraic Geometry of Conics
TLDR
This expository paper describes the solutions to several enumerative problems involving conies, including Steiner's problem, and uses these problems to introduce and demonstrate several of the key ideas and tools of algebraic geometry.
Some Non-Unimodal Level Algebras
In 2005, building on his own recent work and that of F. Zanello, A. Iarrobino discovered some constructions that, he conjectured, would yield level algebras with non-unimodal Hilbert functions. This

References

SHOWING 1-10 OF 15 REFERENCES
Algebraic Geometry
Introduction to Algebraic Geometry.By Serge Lang. Pp. xi + 260. (Addison–Wesley: Reading, Massachusetts, 1972.)
Introduction to Grothendieck Duality Theory
Preface.- Study of ?X.- Completions, primary decomposition and length.- Depth and dimension.- Duality theorems.- Flat morphisms.- Etale morphisms.- Smooth morphisms.- Curves.
Commutative Algebra: with a View Toward Algebraic Geometry
Introduction.- Elementary Definitions.- I Basic Constructions.- II Dimension Theory.- III Homological Methods.- Appendices.- Hints and Solutions for Selected Exercises.- References.- Index of
A standard basis approach to syzygies of canonical curves.
Let C be a smooth projective curve of genus g defined over C and consider the canonical map cpK:C -> P'-* = P(H»(C9a>c)). is an embedding unless C is hyperelliptic. Moreover by a classical result of
Syzygies of Abelian and Bielliptic Surfaces in ℙ4
So far only six families of smooth irregular surfaces are known to exist in P^4 (up to pullbacks by suitable finite covers of P^4). These are the elliptic quintic scrolls, the minimal abelian and
On Petri's analysis of the linear system of quadrics through a canonical curve
where S*H°(I2) denote the symmetric algebra of H°(f2), is surjective. (2) The kernel I of ~p is generated by its elements of degree 2 and of degree 3. (3) I is generated by its elements of degree 2
Syzygies of canonical curves and special linear series
0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 1. Line Bundles on Scrolls . . . . . . . . . . . . . . . . . . . . . . . 109 2. Scrolls and Pencils . . . . . . . . . . . . .
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