Number of defining equations and local cohomology

Name: Number of defining equations and local cohomology
Start: 2025-02-24T16:00:00+00:00
End: 2025-02-24T17:15:00+00:00
Location: Science Center 507

COLLOQUIUMS: SPECIAL COLLOQUIUM, COLLOQUIUMS

When: February 24, 2025

4:00 pm - 5:15 pm

Where: Science Center 507

Address: 1 Oxford Street, Cambridge, MA 02138, United States

Speaker: EMILY WITT - University of Kansas

An algebraic variety is the set of solutions to a system of polynomial equations. Examples include the unit circle x^2+y^2 = 1, a cone such as x^2+y^2-z^2 = 0, and its conic sections obtained by intersecting this cone with a plane.

Consider the system of four equations in four variables given by xu = xv = yu = yv = 0. Is it possible to realize this as the set of solutions to a system of three equations in these same variables? Surprisingly, this type of question can be very hard to answer!

We describe a tool called “local cohomology” that helps us understand the number of equations needed to define a variety. We also discuss other applications of this tool, including connectedness properties of varieties.

There will be a reception in the Math Common Room, Science Center, 4th Floor, immediately following the talk.