Finite sets of points in projective space are fundamental objects in algebraic geometry and commutative algebra. From an algebraic perspective, they correspond to zero-dimensional radical ideals. Despite their apparent simplicity, finite sets of points encode rich combinatorial and geometric information. Their vanishing ideals I(X) capture all polynomials that vanish at the points, while the Hilbert function of I(X) describes the dimension of each graded piece of the coordinate ring, reflecting the distribution and configuration of the points. In the minimal free resolution of I(X) we can see the structure of the syzygies among the generators. The study of symbolic powers I(X)(m) versus ordinary powers I(X)m highlights the interaction between algebraic and geometric multiplicities. These algebraic invariants allow us to understand classical combinatorial configurations such as grids, star configurations of points, Fermat arrangements, and arithmetically Cohen-Macaulay (ACM for short) sets of points in (multi)projective spaces, just to cite some of them. Furthermore, they provide tools for applications in interpolation, coding theory, and algebraic statistics. The course emphasizes algorithmic and computational approaches, showing how modern techniques in commutative algebra can be applied to compute these invariants effectively.

The course consists of 5 lectures.

Lecture 1: Points and Vanishing Ideals

Interpolation, construction of I(X), basic examples.

Lecture 2: Gröbner Bases and Separators

Computation of Gröbner bases for zero-dimensional ideals, standard monomials, Lagrange interpolation, separators.

Lecture 3: Hilbert Functions and Fat Points

Hilbert functions, minimal free resolutions, multiplicities.

Lecture 4: Symbolic Powers and Asymptotic Invariants

Symbolic vs. ordinary powers of I(X), containment problems, Waldschmidt constants, resurgence.

Lecture 5: Applications and Open Problems

Application to algebraic statistics, star and Fermat configurations, algebraic coding theory, open research directions.

References

• E. Guardo, E. and A. Van Tuyl, Arithmetically Cohen-Macaulay Sets of Points in P^1 × P^1. Springer Briefs in Math., Springer-Verlag, Berlin 2015
• D. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms (4th ed.), Springer-Verlag, Berlin 2015.

Domains: Computational Algebraic Geometry

Sessions:

• Elena Guardo:  5 sessions * 75 minutes
   

Exercise Sessions for Course 2:

The exercise classes for Course 2 provide explicitly calculated examples and introduce various concrete techniques for computing the objects explain in the main course.

• Exercise Session 1: Computing the vanishing ideal I(X)
• Exercise Session 2: Construction of separators for given points, computation of the separator degrees
• Exercise Session 3: Compute Hilbert functions and minimal generators for small sets of (fat) points
• Exercise Session 4: Compute powers and symbolic powers of I(X) for small configurations of points
• Exercise Session 5: Discuss the containment conjectures, compute some algebraic codes

Domains: Computational Algebraic Geometry

Sessions:

• Ngoc Long Le: 5 sessions * 75 minutes

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