BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Sabre//Sabre VObject 4.5.8//EN CALSCALE:GREGORIAN BEGIN:VTIMEZONE TZID:Europe/Zurich X-LIC-LOCATION:Europe/Zurich TZURL:http://tzurl.org/zoneinfo/Europe/Zurich BEGIN:DAYLIGHT TZOFFSETFROM:+0100 TZOFFSETTO:+0200 TZNAME:CEST DTSTART:19810329T020000 RRULE:FREQ=YEARLY;BYMONTH=3;BYDAY=-1SU END:DAYLIGHT BEGIN:STANDARD TZOFFSETFROM:+0200 TZOFFSETTO:+0100 TZNAME:CET DTSTART:19961027T030000 RRULE:FREQ=YEARLY;BYMONTH=10;BYDAY=-1SU END:STANDARD END:VTIMEZONE BEGIN:VEVENT UID:news1982@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20260211T131820 DTSTART;TZID=Europe/Zurich:20260217T103000 SUMMARY:Seminar Algebra and Geometry: Michela Artebani (Universidad de Conc epción) DESCRIPTION:The classification of  Mori dream spaces - equivalently\, norm al projective varieties with finitely generated Cox ring - is closely tied to positivity properties of the anticanonical class. In particular\, tori c and Fano varieties provide fundamental classes of examples with finitely generated Cox rings. In contrast\, Calabi-Yau varieties lie at the bounda ry of positivity: there is no general criterion deciding when they are Mor i dream spaces\, and their birational geometry (and birational automorphis m groups) can be remarkably rich.\\r\\nIn this talk we focus on Calabi--Ya u varieties $X$ arising as  general anticanonical hypersurfaces of smooth toric Fano varieties $Z$. Our results are formulated in terms of primitiv e pairs of the anticanonical polytope of $Z$\, which is a smooth reflexive polytope. We present two complementary theorems. The first result provide s sufficient combinatorial conditions on primitive pairs ensuring that $X$ is a Mori dream space\, and it yields an explicit presentation of the Cox ring $R(X)$ in terms of $R(Z)$  and the defining equation of $X$. The se cond result goes in the opposite direction: the existence of certain relat ions among primitive pairs forces $\\mathrm{Bir}(X)$ to be infinite\, and hence $X$ cannot be a Mori dream space. The proof of the first theorem bui lds on the approach of Herrera-Laface-Ugaglia on Cox rings of embedded var ieties\, while the second generalizes ideas of Kawamata and Ottem for anti canonical hypersurfaces in products of projective spaces. As an applicatio n\, we obtain a complete classification of Mori dream Calabi-Yau hypersurf aces in dimensions $2$ and $3$. In particular\, for these hypersurfaces th ere is a sharp dichotomy:  either $R(X)$ is finitely generated  or $\\ma thrm{Bir}(X)$ is infinite.\\r\\nThis is joint work with Antonio Laface and Luca Ugaglia. X-ALT-DESC:

The classification of  \;Mori dream spaces - equivalently \, normal projective varieties with finitely generated Cox ring - is close ly tied to positivity properties of the anticanonical class. In particular \, toric and Fano varieties provide fundamental classes of examples with f initely generated Cox rings. In contrast\, Calabi-Yau varieties lie at the boundary of positivity: there is no general criterion deciding when they are Mori dream spaces\, and their birational geometry (and birational auto morphism groups) can be remarkably rich.

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In this talk we focus on Calabi--Yau varieties $X$ arising as  \;general anticanonical hypersur faces of smooth toric Fano varieties $Z$. Our results are formulated in te rms of primitive pairs of the anticanonical polytope of $Z$\, which is a s mooth reflexive polytope. We present two complementary theorems. The first result provides sufficient combinatorial conditions on primitive pairs en suring that $X$ is a Mori dream space\, and it yields an explicit presenta tion of the Cox ring $R(X)$ in terms of $R(Z)$  \;and the defining equ ation of $X$. The second result goes in the opposite direction: the existe nce of certain relations among primitive pairs forces $\\mathrm{Bir}(X)$ t o be infinite\, and hence $X$ cannot be a Mori dream space. The proof of t he first theorem builds on the approach of Herrera-Laface-Ugaglia on Cox r ings of embedded varieties\, while the second generalizes ideas of Kawamat a and Ottem for anticanonical hypersurfaces in products of projective spac es. As an application\, we obtain a complete classification of Mori dream Calabi-Yau hypersurfaces in dimensions $2$ and $3$. In particular\, for th ese hypersurfaces there is a sharp dichotomy:  \;either $R(X)$ is fini tely generated  \;or $\\mathrm{Bir}(X)$ is infinite.

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This is j oint work with Antonio Laface and Luca Ugaglia.

DTEND;TZID=Europe/Zurich:20260217T120000 END:VEVENT END:VCALENDAR