Baby Algebraic Geometry Seminar

Fall 2016
Mondays at 4:00pm
Location SC 310

Archive from previous years: Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, and Spring 2016.

Below are some ideas for talks. You may want to look at Izzet Coskun's list (with lots of references) or Johan de Jong's seminar page for more topics.

Pretalks

We can have pretalks for the algebraic geometry seminar. That is, talks explaining the background material for the talk in the main seminar. This is immensely helpful. If you see a talk lined up in the main seminar about something dear to you, do consider giving a pretalk.

Foundational Topics

  1. GAGA - basically, analytic geometry in projective space is algebraic. "Géométrie algébraique et géométrie analytique" by Serre, http://www.numdam.org/numdam-bin/item?id=AIF_1956__6__1_0.
  2. Blow ups - they are much more complicated than you would think. "Seven short stories on blow ups and resolutions" by Hauser, http://gokovagt.org/proceedings/2005/ggt05-hauser.pdf.
  3. Serre duality - make it as concrete as possible.
  4. Cohomology and base change - there is an excellent exposition in Mumford's "Abelian Varieties".
  5. Interesting counterexamples.

Curves

  1. Limit linear series - how to extend the notion of linear series on smooth curves to singular curves? "Limit linear series: basic theory" by Eisenbud, Harris, http://www.springerlink.com/content/p0418455005t2pwq/.
  2. Nonreduced structures on P1 - ribbons, carpets etc. "Ribbons and their canonical embeddings" by Eisenbud, http://www.msri.org/~de/papers/pdfs/1995-001.pdf.

Moduli Spaces

  1. Kontsevich's formula for counting rational curves and introduction to quantum cohomology. "Notes on stable maps and quantum cohomology" by Fulton, Pandharipande, http://arxiv.org/abs/alg-geom/9608011.
  2. The effective cone of the Kontsevich spaces. "The effective cone of the Kontsevich moduli space" by Coskun, Harris, Starr, http://www.math.sunysb.edu/~jstarr/papers/reveff.pdf.
  3. Murphy's law in algebraic geometry - this makes precise and proves the following remark of Joe Harris - "There is no geometric possibility so horrible that it doesn't appear on a Hilbert scheme." "Murphy's law in algebraic geometry" by Ravi Vakil, http://math.stanford.edu/~vakil/files/Mjul0705.pdf.
  4. Anything about the moduli of higher dimensional varieties.
  5. Anything about Abelian varieties and their moduli space. The standard reference is "Abelian Varieties" by Mumford. Also see the book "Complex Abelian Varieties" by Birkenhake.

Higher Dimensional Geometry

  1. Rationality, uniruledness, rational connectivity - a concrete topic related to these areas or a general overview. Two good references are the books "Higher dimensional algebraic geometry" by Debarre and "Rational and nearly rational varieties" by Kollár, Smith, Corti.
  2. The cone and contraction theorems - a nice characterization of birational morphisms. "Higher dimensional algebraic geometry" by Debarre or "Rational and nearly rational varieties" by Kollár, Smith, Corti or "Birational geometry of algebraic varieties" by Kollár, Mori.
  3. Flips and flops - these are interesting birational phenomenon occurring in dimension 3 and higher. See "Flips and flops" by McKernan, http://math.mit.edu/~mckernan/Papers/faf.pdf.

Topology/Geometry

  1. The fundamental group of the complement of a curve in P2 - this is related in an intriguing way to the singularities of the curve. "On the problem of existence of algebraic functions of two variables possessing a given branch curve" by Zariski, http://www.jstor.org/pss/2370712.
  2. "The topology of normal singularities of an algebraic surface and a criterion for simplicity." by Mumford, http://www.springerlink.com/content/b18270328x614668/.

Other Interesting Topics

  1. Gale transform - tuples of point in Pn are related in an interesting way to tuples of points in Pm. "The projective geometry of the Gale transform" by Eisenbud, http://arxiv.org/pdf/math/9807127.
  2. Borel-Weil-Bott theorem - representations of certain algebraic groups are related to sections of vector bundles on a projective variety.
  3. Vector bundles on projective spaces - when do vector bundles on Pn split? How do their restrictions to a line change as we move the line? See the book "Vector Bundles on Complex Projective Spaces" by Okonek C., Schneider M., Spindler H..
  4. Computational techniques in algebraic geometry - Gröbner bases etc.