We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth, irreducible, and non-degenerate curve of degree d and genus g in $\mathbb{P}^r.$ In this article, we study $\mathcal{H}_{15,g,5}$ for every possible genus g and determine when it is irreducible. We also study the moduli map $\mathcal{H}_{15,g,5}\rightarrow\mathcal{M}_g$ and several key properties such as gonality of a general element as well as characterizing smooth elements of each component.

Here we classify the weakly uniform rank two vector bundles on multiprojective spaces. Moreover, we show that every rank r>2 weakly uniform vector bundle with splitting type a1,1=⋯=ar,s=0 is trivial and every rank r>2 uniform vector bundle with splitting type a1>⋯>ar splits.

Let X be a smooth projective surface with q(X) = 0 defined over R and M(X;r;c1;c2;H) the moduli space of H-stable rank r vector bundles on X with Chern classes c1 and c2. Assume either r = 3 and X(R) connected or r = 3 and X(R) =ø or r=2 and X(R) = ø. We prove that quite often M is connected.

A line bundle, L, on a smooth, connected projective surface, S, is defined [7] to be k-very ample for a non-negative integer, k, if given any 0-dimensional sub-scheme with length , it follows that the restriction map is onto. L is 1-very ample (respectively 0-very ample) if and only if L is very ample (respectively spanned at all points by global sections). For a smooth surface, S, embedded in projective space by | L | where L is very ample, L being k-very ample is equivalent to there being no k-secant Pk−1 to S containing ≥ k + 1 points of S.