More Like this

1. Cohomological Blowups of Graded Artinian Gorenstein Algebras along Surjective Maps

   https://doi.org/10.1093/imrn/rnac002  

   Iarrobino, Anthony; Macias Marques, Pedro; McDaniel, Chris; Seceleanu, Alexandra; Watanabe, Junzo (February 2022, International Mathematics Research Notices)

   Abstract We introduce the cohomological blowup of a graded Artinian Gorenstein algebra along a surjective map, which we term BUG (blowup Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blowup of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are complete intersections. We have included many examples throughout this manuscript. 

   more » « less
2. Positivity and nonstandard graded Betti numbers

   https://doi.org/10.1112/blms.12917  

   Brown, Michael K; Erman, Daniel (January 2024, Bulletin of the London Mathematical Society)

   Abstract A foundational principle in the study of modules over standard graded polynomial rings is that geometric positivity conditions imply vanishing of Betti numbers. The main goal of this paper is to determine the extent to which this principle extends to the nonstandard ‐graded case. In this setting, the classical arguments break down, and the results become much more nuanced. We introduce a new notion of Castelnuovo–Mumford regularity and employ exterior algebra techniques to control the shapes of nonstandard ‐graded minimal free resolutions. Our main result reveals a unique feature in the nonstandard ‐graded case: the possible degrees of the syzygies of a graded module in this setting are controlled not only by its regularity, but also by its depth. As an application of our main result, we show that given a simplicial projective toric variety and a module over its coordinate ring, the multigraded Betti numbers of are contained in a particular polytope when satisfies an appropriate positivity condition. 

   more » « less
3. Lim Ulrich sequences and Boij-Söderberg cones

   https://doi.org/10.1017/fms.2023.108  

   Iyengar, Srikanth B; Ma, Linquan; Walker, Mark E (January 2023, Forum of Mathematics, Sigma)

   Abstract This paper extends the results of Boij, Eisenbud, Erman, Schreyer and Söderberg on the structure of Betti cones of finitely generated graded modules and finite free complexes over polynomial rings, to all finitely generated graded rings admitting linear Noether normalizations. The key new input is the existence of lim Ulrich sequences of graded modules over such rings. 

   more » « less
4. Geometric Aspects of the Jacobian of a Hyperplane Arrangement

   https://doi.org/10.1093/imrn/rnaf172  

   DiPasquale, Michael; Sidman, Jessica; Traves, Will (July 2025, International Mathematics Research Notices)

   Abstract An embedding of the complete bipartite graph $$K_{3,3}$$ in $$\mathbb{P}^{2}$$ gives rise to both a line arrangement and a bar-and-joint framework. For a generic placement of the six vertices, the graded Betti numbers of the logarithmic module of derivations of the line arrangement are constant, but an important example due to Ziegler shows that the graded Betti numbers are different when the points lie on a conic. Similarly, in rigidity theory a generic embedding of $$K_{3,3}$$ in the plane is an infinitesimally rigid bar-and-joint framework, but the framework is infinitesimally flexible when the points lie on a conic. We develop the theory of weak perspective representations of hyperplane arrangements to formalize and generalize the striking connection between hyperplane arrangements and rigidity theory that the example above suggests. In characteristic zero we show that there is a one-to-one correspondence between weak perspective representations of a hyperplane arrangement and polynomials of minimal degree in certain saturations of the Jacobian ideal of the arrangement, providing a connection to algebra. In this setting we can use duality theorems to explain how rigidity theory is reflected in the graded Betti numbers of the module of logarithmic derivations of a line arrangement. 

   more » « less
5. Stable behavior of Frobenius powers over a general hypersurface

   https://doi.org/10.1090/btran/199  

   Miller, Claudia; Rahmati, Hamidreza; R_G, Rebecca (November 2024, Transactions of the American Mathematical Society, Series B)

   The main goal of this paper is to prove, in positive characteristic $p$, stability behavior for the graded Betti numbers in the periodic tails of the minimal resolutions of Frobenius powers of the homogeneous maximal ideals for very general choices of hypersurface in three variables whose degree has the opposite parity to that of $p$. We also find some of the structure of the matrix factorization giving the resolution. We achieve this by developing a method for obtaining the degrees of the generators of the defining ideal of an $\mathfrak{c}$-compressed Gorenstein Artinian graded algebra from its socle degree, where $\mathfrak{c}$is a Frobenius power of the homogeneous maximal ideal. As an application, we also obtain the Hilbert–Kunz function of the hypersurface ring, as well as the Castelnuovo–Mumford regularity of the quotients by Frobenius powers of the homogeneous maximal ideal. 

   more » « less