Grobner Basis and Surfaces Grobner basis are really good at describing polynomial systems of equations with 0-dimensional zero sets. In a sense, Grobner basis yields a better/simplified description of such systems because the Grobner polynomials (in appropriate monomial order) let one explicitly compute the zeros. 
However, suppose the ideal $I\subseteq k[x_1,\ldots,x_n]$ has a positive dimensional zero set, such as a curve or surface. What can Grobner basis tell you about these zero sets? I'm particularly interested in the curves and surfaces case (as I am intersecting the flat 3-torus with various varieties). 
 A: Chapter $15$ of Eisenbud's Commutative Algebra: with a View Toward Algebraic Geometry contains many applications of Gröbner bases, and many of these are not special to $0$-dimensional ideals.  An initial list can be found in $\S15.1$ (p. $318$), while a more detailed treatment occurs in $\S15.10$ (p. $355$).  I will list a couple here.  Let $k$ be a field and $S = k[x_1, \ldots, x_r]$ be a polynomial ring.
Ideal membership: The most fundamental problem that Gröbner bases solve is that of ideal membership: given an ideal $I \trianglelefteq S$ and $f \in S$, how can we tell if $f \in I$?  One can show that the remainder of $f$ upon division by a Groebner basis is unique, so we have that $f \in I$ iff its remainder is $0$, just as in the single variable case.
The geometric significance is that if $S/I$ is the coordinate ring of an irreducible affine algebraic set (with $I = \sqrt{I}$), a polynomial $f \in S$ induces the zero function on $V$ iff $f \in I$.  So this allows us to determine when two polynomials $f,g \in S$ induce the same regular function when restricted to $V$: $f = g$ on $V$ iff $f-g \in I$.
Hilbert functions: Given a projective variety $V$, let $M$ be its homogeneous coordinate ring, which is a graded module over $S$.  Let $M_s$ denote the $s^\text{th}$ graded piece of $M$.  Then $H_M(s) := \dim_k(M_s)$ is the Hilbert function of $M$ (or of $V$).  Gröbner bases provide a simple way of computing the Hilbert function that only requires considering monomial ideals.  (Here $\DeclareMathOperator{\init}{in} \init(N)$ is the initial submodule with respect to some monomial ordering.)
Theorem 15.26. Let $M$ be a finitely generated graded $S$-module, given by generators and relations as $M = F/N$, where $F$ is a free module with a homogeneous basis and $N$ is a submodule generated by homogeneous elements.  The Hilbert function of $M$ is the same as the Hilbert function of $F/\init(N)$.
For sufficiently large integers the Hilbert function agrees with a polynomial with rational coefficients called the Hilbert polynomial.  The Hilbert polynomial carries several pieces of information about a projective variety, such as its dimension, degree, and arithmetic genus.  Some other information that can be extracted from the Hilbert polynomial is mentioned in this MO thread.
Primary decomposition: From a Gröbner basis, one can compute the primary decomposition of an ideal $I \trianglelefteq S$, i.e., one can write $I = Q_1 \cap \cdots \cap Q_m$ where the $Q_i$ are primary.  In geometric terms, this allows one to decompose an affine algebraic set into irreducible varieties since
$$
\mathbb{V}(I) = \mathbb{V}(Q_1 \cap \cdots \cap Q_m) = \mathbb{V}(Q_1) \cup \cdots \cup \mathbb{V}(Q_m) \, .
$$
(Technically, primary decomposition carries even more information than the classical geometric picture, since $\mathbb{V}(I) = \mathbb{V}(\sqrt{I})$ if $k$ is algebraically closed.)
Eisenbud includes quite a few other applications of Gröbner bases, too, such as computation of syzygies, free resolutions, and important functors like $\text{Hom}, \text{Ext}$, and $\text{Tor}$.
