Title: On the degrees of minimal generators of Pinched Veronese algebras

Abstract: In this talk, we consider the homogeneous coordinate rings A(PV(n,d,s)) ≅ K[Pₙ,𝑑,𝑠] of pinched Veronese varieties, i.e., monomial projections PV(n,d,s) of Veronese varieties parameterized by subsets Pₙ,𝑑,𝑠 which contain all monomials of degree d in n+1 variables supported in at most s variables.

Our goal is to determine an upper bound for the degrees of the generators of the ideal of the pinched Veronese varieties. As a particular case, we study when K[Pₙ,𝑑,𝑠] is a quadratic algebra and address the following question:

Is it true that K[Pₙ,𝑑,𝑠] is quadratic if and only if s ≥ ⌈(n+2)/2⌉, with a few sporadic exceptions?

This is a joint work with Simone Marchesi (Universidad de Barcelona) and Rosa Maria Miró-Roig (Universidad de Barcelona).

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Title: Compressed Artinian Gorenstein Algebras

Abstract: In this work, we determine a specific family of forms of degree d which turns out to be the Macaulay dual generators of standard graded compressed Artinian Gorenstein algebras A_F satisfying the Weak Lefschetz Property (WLP, for short) and the Strong Lefschetz property (SLP, for short).

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Title: On the Torelli Theorem for graphs and stable curves

Abstract: The classical Torelli theorem states that a smooth curve can be recovered from its polarized Jacobian. In this talk, we will discuss the extensions of this theorem to stable curves and their dual graphs, and its dependence on the concept of compactified Jacobians.

Title: A combinatorial point of view for the nested Hilbert schemes of points on affine spaces

Abstract: In this talk we realize the nested Hilbert scheme of points on affine varieties as quiver varieties of  enhanced ADHM quivers. This construction generalizes previous work of Jardim, von Flach and Lanza about the nested Hilbert schemes of points on A2 and also a work of Henni and Jardim, about the Hilbert schemes of points on An, for n ≥ 3. The correspondence is based on a monadic approach and represents one more connection between Algebraic Geometry and Representation Theory.

Title: Stability Conditions for Coherent Systems on Integral Curves

Abstract: In this talk, we start by recalling the concepts we will use, namely,  (a) the categorical notion of stability; (b) the definition of coherent systems on algebraic varieties; and (c) the differences coming from allowing singularities on the curves we deal with. As for the results, we define a three-parameter family of pre-stability conditions in the derived category of coherent systems on curves using tilting methods, and we then investigate when these conditions qualify as true stability conditions. Additionally, we examine the semistability of specific objects under these conditions, such as torsion, free, and complete tilted systems, without relying on the support property. At the end, we discuss some possible applications of the results we obtained. It is a joint work with Marcos Jardim and Leonardo Roa-Leguizamón.

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