Oberwolfach Seminar on Computational Algebraic Geometry
Examples and Applications of Computational Algebraic Geometry
Frank Sottile

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General Schedule I) Toric Varieties 3.5 lectures II) Grassmannians 1.5 lectures Current Exercises from Sottile. Sottile's outline of Frank Schreyer's First Lecture. Sottile's outline of Christoph Lossen's First Lecture.

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Lecture 1: Toric Ideals

• Motivation: solving equations
• Affine toric variety
• Elementary properties of toric ideals
• Generation of toric ideals

Lecture 2: Flatness and Gröbner Degeneration

• Definition and motivation for flat family
• Flat families over smooth curves
• Degenerations to initial ideals of weight orders

Lecture 3: Toric deformations to initial ideals

• Gröbner degenerations
• Initial ideals of toric ideals
• Polyhedral subdivisions and initial ideals
• Kouchnirenko's Theorem

Lecture 4: Reality and the Viro construction.

• Hilbert Polynomial of a Toric ideal, Kouchnirenko's Theorem
• Sturmfels Reality Theorem, Kouchnirenko's Theorem
• The Viro Construction

        Grassmann Varieties

• Motivation: 4 lines in space and ramification of linear series
• The Grassmann Variety

Lecture 5: Ideal of the Grassmann

• The Plücker ideal
• Degree of Grassmannian

Topics skipped:

• The Cox coordinate ring
• Real Toric varieties and the moment map
• Real Dergree of the Grassmannian and Shapiros' conjecture

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Last modified: 1 December 2003