Related papers: Geometry of arithmetically Gorenstein curves in ${…

The quantum cohomology algebra of a projective manifold X is the cohomology H(X,Q) endowed with a different algebra structure, which takes into account the geometry of rational curves in X. We show that this algebra takes a remarkably…

Planar graphs can be represented as intersection graphs of different types of geometric objects in the plane, e.g., circles (Koebe, 1936), line segments (Chalopin \& Gon{\c{c}}alves, 2009), \textsc{L}-shapes (Gon{\c{c}}alves et al, 2018).…

An integral geometric curvature is defined as the index expectation K(x) = E[i(x)] if a probability measure m is given on vector fields on a Riemannian manifold or on a finite simple graph. Such curvatures are local, satisfy Gauss-Bonnet…

We classify Cohen-Macaulay graphs of girth at least $5$ and planar Gorenstein graphs of girth at least $4$. Moreover, such graphs are also vertex decomposable.

We consider geometrical problems on Gorenstein hypersurface orbifolds of dimension $n \geq 4$ through the theory of Hilbert scheme of group orbits. For a linear special group $G$ acting on $\CZ^n$, we study the $G$-Hilbert scheme,…

Let $X$ be the blow up of $\mathbb{P}^2$ at $r$ general points $p_1,\ldots,p_r \in \mathbb{P}^2$. We study line bundles on $X$ given by plane curves of degree $d$ passing through $p_i$ with multiplicity $m_i$. We establish conditions for…

We study the geometry of surfaces in $\mathbb{R}^{4}$ with corank $1$ singularities. For such surfaces the singularities are isolated and at each point we define the curvature parabola in the normal space. This curve codifies all the second…

We develop an algebraic version of Cartan method of equivalence or an analog of Tanaka prolongation for the (extrinsic) geometry of curves of flags of a vector space $W$ with respect to the action of a subgroup $G$ of the $GL(W)$. Under…

In this paper, we study the curve cone of an almost complex $4$-manifold which is tamed by a symplectic form. In particular, we prove the cone theorem as in Mori theory for all such manifolds using the Seiberg-Witten theory. For small…

After defining generalizations of the notions of covariant derivatives and geodesics from Riemannian geometry for reductive Cartan geometries in general, various results for reductive Cartan geometries analogous to important elementary…

We study geometrical properties of maximal curves having classical Weierstrass gaps.

We show that any tetragonal Gorenstein integral curve is a complete intersection in its respective $3$-fold rational normal scroll S, implying that the normal sheaf on $C$ embedded in S, and in $\mathbb{P}^{g-1}$ as well, is unstable for…

Codimension 2 complete intersections in P^N have a natural parameter space \bar{H}: a projective bundle over a projective space given by the choice of the lower degree equation and of the higher degree equation up to a multiple of the…

The modularity of elliptic curves always intrigues number theorists. Recently, Thorne had proved a marvelous result that for a prime $ p $, every elliptic curve defined over a $ p $-cyclotomic extension of $ \mathbb{Q} $ is modular. The…

We present methods to construct interesting surfaces of general type via $\mathbb{Q}$-Gorenstein smoothing of a singular surface obtained from an elliptic surface. By applying our methods to special Enriques surfaces, we construct new…

This paper provides both the theoretical results and numerical calculations of global solution curves, by continuation in global parameters. Each point on the solution curves is computed directly as the global parameter is varied, so that…

We extend the result of Lisini (Calc Var Partial Differ Equ 28:85-120, 2007) on the superposition principle for absolutely continuous curves in $p$-Wasserstein spaces to the special case of $p=1$. In contrast to the case of $p>1$, it is not…

Special curves and their characterizations are one of the main area of mathematicians and physicians. As a special curve we will mainly focus on Mannheim curve which has the following relation: k1={\beta}(k1^2+k2^) where k1 and k2 are…

Let $K_d$ be the complete metric graph on $d$ vertices. We compute the gonality of graphs obtained from $K_d$ by omitting edges forming a $K_h$, or general configurations of at most $d-2$ edges. We also investigate if these graphs can be…

We describe the arrangement of all Galois lines for the Giulietti--Korchm\'{a}ros curve in the projective $3$-space. As an application, we determine the set of all Galois points for a plane model of the GK curve. This curve possesses many…