BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Victor Batyrev (University of Tübingen) DTSTART:20210611T120000Z DTEND:20210611T130000Z DTSTAMP:20260404T094702Z UID:ToricDeg/1 DESCRIPTION:Title: Variations on the theme of classical discriminant\nby Victor B atyrev (University of Tübingen) as part of Toric Degenerations\n\n\nAbstr act\nThe classical discriminant $\\Delta_n(f)$ of a degree $n$ polynomial $f(x)$ is an irreducible homogeneous polynomial of degree $2n-2$ on the co efficients $a_0\, \\ldots\, a_n$ of $f$ that vanishes if and only if $f$ has a multiple zero. I will explain a tropical proof of the theorem of Gel fand\, Kapranov and Zelevinsky (1990) that identifies the Newton polytope $P_n$ of $\\Delta_n$ with a $(n-1)$-dimensional combinatorial cube obtain ed from the classical root system of type $A_{n-1}$. Recently Mikhalkin an d Tsikh (2017) discovered a nice factorization property for truncations of $\\Delta_n$ with respect to facets $\\Gamma_i$ of $P_n$ containing the ve rtex $v_0 \\in P_n$ corresponding to the monomial $a_1^2 \\cdots a_{n-1}^ 2 \\in \\Delta_n$. I will give a GKZ-proof of this property and show its c onnection to the boundary stata in the $(n-1)$-dimensional toric Losev-Man in moduli space $\\overline{L_n}$. Some variations on the above statements will be discussed in connection to the toric moduli space associated with the root system of type $B_n$ and the mirror symmetry for $3$-dimensional cyclic quotient singularities ${\\mathbb C}^3/\\mu_{2n+1}$.\n LOCATION:https://stable.researchseminars.org/talk/ToricDeg/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Lara Bossinger (UNAM Oaxaca) DTSTART:20210611T131500Z DTEND:20210611T141500Z DTSTAMP:20260404T094702Z UID:ToricDeg/2 DESCRIPTION:Title: Newton--Okounkov bodies for cluster varieties\nby Lara Bossing er (UNAM Oaxaca) as part of Toric Degenerations\n\n\nAbstract\nCluster var ieties are schemes glued from algebraic tori. Just as tori themselves\, th ey come in dual pairs and it is good to think of them as generalizing tori . Just as compactifications of tori give rise to interesting varieties\, ( partial) compactifications of cluster varieties include examples such as G rassmannians\, partial flag varieties or configurations spaces. A few year s ago Gross--Hacking--Keel--Kontsevich developed a mirror symmetry inspire d program for cluster varieties. I will explain how their tools can be use d to obtain valuations and Newton--Okounkov bodies for their (partial) com pactifications. The rich structure of cluster varieties however can be exp loited even further in this context which leads us to an intrinsic definit ion of a Newton--Okounkov body.\nThe theory of cluster varieties interacts beautifully with representation theory and algebraic groups. I will exhib it this connection by comparing GHKK's technology with known mirror symmet ry constructions such as those by Givental\, Baytev--Ciocan-Fontanini--Kim --van Straten\, Rietsch and Marsh--Rietsch.\n LOCATION:https://stable.researchseminars.org/talk/ToricDeg/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Chris Manon (University of Kentucky) DTSTART:20210611T143000Z DTEND:20210611T153000Z DTSTAMP:20260404T094702Z UID:ToricDeg/3 DESCRIPTION:Title: When is a (projectivized) toric vector bundle a Mori dream space?< /a>\nby Chris Manon (University of Kentucky) as part of Toric Degeneration s\n\n\nAbstract\nLike toric varieties\, toric vector bundles are a rich cl ass of varieties which admit a combinatorial description. Following the c lassification due to Klyachko\, a toric vector bundle is captured by a sub space arrangement decorated by toric data. This makes toric vector bundle s an accessible test-bed for concepts from algebraic geometry. Along the se lines\, Hering\, Payne\, and Mustata asked if the projectivization of a toric vector bundle is always a Mori dream space. Suess and Hausen\, an d Gonzales showed that the answer is "yes" for tangent bundles of smooth\, projective toric varieties\, and rank 2 vector bundles\, respectively. T hen Hering\, Payne\, Gonzales\, and Suess showed the answer in general mus t be "no" by constructing an elegant relationship between toric vector bun dles and various blow-ups of projective spaces\, in particular the blow-up s of general arrangements of points studied by Castravet\, Tevelev and Muk ai. In this talk I'll review some of these results\, and then give a new description of toric vector bundles by tropical information. This descrip tion allows us to characterize the Mori dream space property in terms of t ropical and algebraic data\, and produce new families of Mori dream spaces indexed by the integral points in a locally closed polyhedral complex. Along the way I'll discuss plenty of examples and some questions. This i s joint work with Kiumars Kaveh.\n LOCATION:https://stable.researchseminars.org/talk/ToricDeg/3/ END:VEVENT END:VCALENDAR