Harvard/MIT Algebraic Geometry Seminar

Spring 2019
Tuesdays at 3 pm

The Harvard/MIT Algebraic Geometry Seminar will alternate between MIT (2-147) and Harvard (Science Center 507).

• Feb 52018 SC 507 Harvard Mihai Fulger, University of Connecticut
  Seshadri constants for bundles abstract±    For line bundles, Seshadri constants are classical local positivity invariants at a point. They detect ampleness in several ways, and they measure jet separation at the point in an asymptotic sense. A generalization of Seshadri constants to higher rank was first studied explicitly by Hacon. In our work, we extend the classical rank one results. We also give three applications. First, we offer a Seshadri characterization of projective space as the only Fano manifold whose tangent bundle has positive Seshadri constant for at least one point. We conjecture that the Fano assumption can be removed, and offer the case of surfaces as evidence. Second, we make partial progress towards a conjecture on the shape of the nef cone of the self product of a very general curve of high genus. Third, we observe that Seshadri constants control jet separation for direct images of pluricanonical sheaves. This is all in joint work with Takumi Murayama.
• Feb 122019 SC 507 Harvard Dan Abramovich, Brown University
  Moduli technique in resolution of singularities abstract±    Semistable reduction, a form of resolution of singularities of families, is often the first step in constructing compactified moduli spaces, and can be used to discover their properties. I will describe work-in-progress with Michael Temkin and Jaroslaw Wlodarczyk in which we prove functorial semistable reduction for families of varieties in characteristic 0, refining work with Karu from 2000. Techniques developed for moduli spaces enter in unexpected ways.
• Feb 192018 2-147 MIT Maksym Fedorchuk, Boston College
  Standard models of low degree del Pezzo fibrations abstract±    A del Pezzo fibration is one of the natural outputs of the Minimal Model Program for threefolds. At the same time, geometry of an arbitrary del Pezzo fibration can be unsatisfying due to the presence of non-integral fibers and terminal singularities of an arbitrarily large index. In 1996, Corti developed a program of constructing `standard models' of del Pezzo fibrations within a fixed birational equivalence class. Standard models enjoy a variety of desired properties, one of which is that all of their fibers are Q-Gorenstein integral del Pezzo surfaces. Corti proved the existence of standard models for del Pezzo fibrations of degree d\geq 2, with the case of d=2 being the most difficult. The case of d=1 remained a conjecture. In 1997, Kollár recast and improved the Corti’s result in degree d=3 using ideas from the Geometric Invariant Theory for cubic surfaces. I will present a generalization of Kollár’s approach in which we develop notions of stability for families of low degree (d\leq 2) del Pezzo fibrations in terms of their Hilbert points (i.e., low degree equations cutting out del Pezzos). A correct choice of stability and a bit of enumerative geometry then leads to (very good) standard models in the sense of Corti. This is a joint work with Hamid Ahmadinezhad and Igor Krylov.
• Feb 262019 SC B10 Harvard Ethan Cotterill, Universidade Federal Fluminense (**Note unusual room**)
  Real inflection points of real linear series on real (hyper)elliptic curves (joint with I. Biswas and C. Garay López) abstract±    According to Plucker's formula, the total inflection of a linear series (L,V) on a complex algebraic curve C is fixed by numerical data, namely the degree of L and the dimension of V. Equipping C and (L,V) with compatible real structures, it is more interesting to ask about the total real inflection of (L,V). The topology of the real inflectionary locus depends in a nontrivial way on the topology of the real locus of C. We study this dependency when C is hyperelliptic and (L,V) is a complete series. We first use a nonarchimedean degeneration to relate the (real) inflection of complete series to the (real) inflection of incomplete series on elliptic curves; we then analyze the real loci of Wronskians along an elliptic curve, and formulate some conjectural quantitative estimates.
• Mar 52019 2-147 MIT Sho Tanimoto, Kumamoto University
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• Mar 122019 SC 507 Harvard Jake Levinson, University of Washington
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• Mar 192019 2-143 MIT Luc Illusie, Université Paris-Sud, Talk 1, 3-4 (**Note two seminar talks and unusual room**)
  The de Rham-Witt complex: review and prospects abstract±    I will recall the historical background, some of the main results, and discuss open questions raised by the new approach of Bhatt, Lurie and Mathew.
• Mar 192019 2-143 MIT Luc Illusie, Université Paris-Sud, Talk 2, 4:30-5:30
  Remarks on the cotangent complex and the Nygaard filtration: DI revisited abstract±    I'll revisit decompositions of de Rham complexes in positive characteristic (Deligne-Illusie), by discussing some relations between the cotangent complex, liftings mod $p^2$, and the de Rham-Witt complex.
• Mar 262019 SC 507 Harvard Changho Han, Harvard University
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• Apr 22019 2-147 MIT Tomasz Szemberg, Pedagogical University of Cracow
  Postulation in projective spaces and unexpected hypersurfaces abstract±    Given a subscheme X in projective space, it is a very classical problem to determine the dimension of the vector space of all homogeneous polynomials of some fixed degree d vanishing along X. Even in the simplest case, when X is supported on a finite set of points in the projective plane, a complete answer is not known and it is subject to open conjectures due to Nagata (1959) and Segre-Harbourne-Gimigliano-Hirschowitz (1964-78). Recently Cook II, Harbourne, Migliore and Nagel observed that the situation becomes even more interesting if one replaces the space of all polynomials of certain degree by its carefully chosen subspaces. I will focus on examples exhibiting these new phenomena. This is based on joint work with Bauer, Malara and Szpond.
• Apr 92019 SC 507 Harvard TBD
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• Apr 162019 2-147 MIT TBD
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• Apr 232019 SC 507 Harvard Kalina Mincheva, Yale University
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• Apr 302019 2-147 MIT TBD
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• May 72019 SC 507 Harvard TBD
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• May 142019 2-147 MIT TBD
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This seminar is organized by Joe Harris (Harvard), Davesh Maulik (MIT), Brooke Ullery (Harvard). This seminar is supported in part by grants from the NSF. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.