Abstract:
The n-dimensional toric hypersurfaces $X$ are affine varieties defined by the unique equation $f=0$ in the algebraic torus $T$ of dimension $n+1$. The main discrete invariant of $X$ is the Newton polytope $P$ of its Laurent polynomial $f$. If the coefficients of $f$ satisfy some explicitly described Zariski open non-degeneracy condition, then many birational properties of the variety $X$ and various geometric invariants of its projective birational models can be found using elementary convex geometry through the polytope $P$. From a pedagogical point of view, toric hypersurfaces allow beginners in algebraic geometry to visually construct a rich set of examples illustrating the general classical theory of algebraic curves and algebraic surfaces. Moreover, relying on examples in dimension 1 and 2, using non-degenerate toric hypersurfaces, it is possible to form additional intuition for new studies and obtaining new results in the general birational classification of varieties of dimension 3 and higher. The purpose of the talk is to introduce listeners to these possibilities and to explain the most optimal way to achieve all points of the Mori program for non-degenerate toric hypersurfaces in arbitrary dimension $n$.
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