Research Projects

In this project, we will investigate the structural and homological properties of certain finitely generated graded algebras over a field associated with some combinatorial and geometric objects, especially in toric geometry. This research mainly focuses on understanding the invariant Castelnuovo-Mumford regularity and a fundamental property, "Green-Lazarsfeld's $N_p$ property." The Castelnuovo-Mumford regularity of a graded algebra $A$, denoted by $reg(A)$, is an extremely important invariant as it measures the complexity of the graded algebra. The $i$-th syzygy module of a graded algebra $A$ can be generated by the elements of degree less or equal to the $reg(A) + i$. Therefore, finding the Castelnuovo-Mumford regularity or giving some bounds on it is very important. Also, it is very important to study the properties of its minimal free resolution to understand the algebra and its syzygies. For a non-negative integer $p$, the algebra $A$ satisfies the property $N_p$ if the $i,i+j+1$-th Betti-number vanishes for all $I \leq p$ and $j \geq 1$. For example, $A$ satisfies $N_1$ if and only if the defining ideal is generated by quadrics. When the property $N_p$ holds for all $p$, then the minimal free resolution of the defining ideal is as nice as possible. The regularity and $N_p$-property of finitely generated graded algebras have been studied extensively in the literature to understand the computational and geometric complexity of the algebra. This project will focus on the toric algebras, especially defined by generic toric ideals, Veronese rings and their subrings, Projective monomial curves, and normal affine semigroups. In [M. J. Nitsche. J. Algebra, 368:345-357, 2012], an explicit formula is given for the Castelnuovo-Mumford regularity of full veronese rings, but there is no formula for the subrings of veronese rings. The projective monomial curves are a special case of subrings of Veronese rings. There is a long-standing conjecture by Eisebud and Goto [D. Eisenbud and S. Goto. J. Algebra, 88(1):89-133, 1984], concerning the regularity of a standard graded algebra $A$ over an algebraically closed field, it says that if $A$ is a domain, then the regularity of $A$ is less or equal to the difference of the multiplicity and the codimension of $A$. This conjecture is not generally true, as McCullough and Peeva gave a counterexample in [J. McCullough and I. Peeva. J. Amer. Math. Soc., 31(2):473-496, 2018], but this is open in affine semigroup rings. We will explore the Castelnuovo-Mumford regularity for the generic toric ideals and explore Eisenbud-Goto's conjecture for semigroup rings whose defining ideal is generic toric. Afterward, we will investigate the $N_p$ property for toric rings and explore Ottaviani-Paoletti's conjecture for the $N_p$ property (it conjectures that for which $p$ and degree $d$ a Veronese ring satisfy $N_p$-property) of Veronese rings with combinatorial approaches.