Toric Surfaces Pcmi Undergraduate Course, July 2008 Jessica Sidman with Help from Jose Gonzalez

An Introduction to Toric Surfaces

Example 1.1. In R, the parabola is the set of all points satisfying y = x. We see that for any value of x, there is a unique value of y that satisfies the equation. In fact, we can parameterize the parabola with a map φ : R→ R where φ(t) = (t, t). For each value that the parameter t assumes we get a point in R on the parabola. For example, φ(−5) = (−5, 25), φ(0) = (0, 0), φ(5) = (5, 25) are all...

Linear determinantal equations for all projective schemes

We prove that every projective embedding of a connected scheme determined by the complete linear series of a sufficiently ample line bundle is defined by the 2×2 minors of a 1-generic matrix of linear forms. Extending the work of Eisenbud-Koh-Stillman for integral curves, we also provide effective descriptions for such determinantally presented ample line bundles on products of projective space...

A Sharp Bound for the Castelnuovo-mumford Regularity of Subspace Arrangements Harm Derksen and Jessica Sidman

Over the past twenty years rapid advances in computational algebraic geometry have generated increasing amounts of interest in quantifying the “complexity” of ideals and modules. For a finitely generated module M over a polynomial ring S = k[x0, . . . , xn] with k an arbitrary field, we say that M is r-regular (in the sense of Castelnuovo and Mumford) if the i-th syzygy module of M is generated...

Syzygies of the Secant Variety of a Curve

We show the secant variety of a linearly normal smooth curve of degree at least 2g + 3 is arithmetically Cohen-Macaulay, and we use this information to study the graded Betti numbers of the secant variety.

Weak immersions of surfaces with L 2 - bounded second fundamental form Lecture 3 PCMI Graduate Summer School 2013