What are some open problems in toric varieties? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T14:55:29Z http://mathoverflow.net/feeds/question/43882 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43882/what-are-some-open-problems-in-toric-varieties What are some open problems in toric varieties? James Davidoff 2010-10-27T21:39:21Z 2011-01-17T16:11:09Z <p>In light of the nice responses to <a href="http://mathoverflow.net/questions/37172/what-are-some-open-problems-in-algebraic-geometry" rel="nofollow">this question</a>, I wonder what are some open problems in the area of toric geometry? In particular, </p> <blockquote> <p>What are some open problems relating to the algebraic combinatorics of toric varieties?</p> </blockquote> <p>and</p> <blockquote> <p>What are some open problems relating to the algebraic geometry of toric varieties?</p> </blockquote> http://mathoverflow.net/questions/43882/what-are-some-open-problems-in-toric-varieties/52313#52313 Answer by J.C. Ottem for What are some open problems in toric varieties? J.C. Ottem 2011-01-17T13:45:34Z 2011-01-17T14:13:24Z <p>The following question was posed by Rikard Bögvad in the paper <a href="http://arxiv.org/abs/alg-geom/9501012" rel="nofollow">On the homogeneous ideal of a projective nonsingular toric variety</a>:</p> <blockquote> <p>Is the toric ideal of a smooth projectively normal toric variety generated by quadrics?</p> </blockquote> <p>This is interesting, since toric ideals have an explicit description. In particular, it is not known if the coordinate ring of a smooth projectively normal toric variety is Koszul. Smoothness is of course essential here, since there are many toric hypersurfaces of degree \$\ge 3\$, e.g., \$x_0^n=x_1 \cdots x_n\$.</p> http://mathoverflow.net/questions/43882/what-are-some-open-problems-in-toric-varieties/52331#52331 Answer by Alexander Duncan for What are some open problems in toric varieties? Alexander Duncan 2011-01-17T16:11:09Z 2011-01-17T16:11:09Z <p>My favourite is Oda's Strong Factorization Conjecture:</p> <blockquote> <p>Can a proper, birational map between smooth toric varieties be factored as a composition of a sequence of smooth toric blow-ups followed by a sequence smooth toric blow-downs?</p> </blockquote> <p>Note that if you are allowed to intermingle the blow-ups and blow-downs (the weak version) it has been proved. In fact, it was proved for general varieties in characteristic 0 <em>using</em> the toric case:</p> <p><a href="http://www.ams.org/journals/jams/2002-15-03/" rel="nofollow">Torification and Factorization of Birational Maps.</a> Abramovich, Karu, Matsuki, Wlodarczyk.</p> <p>A conjectural algorithm for computing toric strong factorizations can be found in the following arXiv article:</p> <p><a href="http://arxiv.org/abs/0911.4693" rel="nofollow">On Oda's Strong Factorization Conjecture.</a> Da Silva, Karu.</p>