UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

March 11, 2008

939 Evans Hall

3:00PM: Every Algebraic Set in n-Space is an intersection of n Hypersurfaces (after Eisenbud and Evans)

Morgan Brown

In 1882, Kronecker showed that any algebraic set in k^n is the intersection of n+1 hypersurfaces. Since then, mathematicians have endeavored to discover the smallest number of necessary hypersurfaces. I will give a short overview of the history of the problem, culminating in Eisenbud and Evans' result that n such surfaces suffice.

5:45PM: Spaces of rational curves on hypersurfaces of high degree

Roya Beheshti

Consider the space of smooth rational curves contained in in a general hypersurfaces of degree d in P^n. When d is at least (n+1)/2, these spaces have irreducible components which are non-uniruled. I will describe the ideas of the proof of this result and its connection with the question of rationality of hypersurfaces. I will also discuss a new proof of a theorem of de Jong-Starr on the spaces of rational curves on general cubic fourfolds.

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