Let $X\subset \mathbb{P}^N$ be an irreducible non-degenerate variety. If the $(h,k)$-Grass\-mann secant variety $G_{h,k}(X)$ of $X$ is not the whole Grassmannian $\mathbb{G}(h,N)$, we have that the singular locus of $G_{h,k}(X)$ contains $G_{h,k-1}(X)$. Moreover, if $X$ is a smooth curve without $(2k+2)$-secant $2k$-space divisors, we obtain the equality $\text{Sing}(G_{h,k}(X))=G_{h,k-1}(X)$.

We study a class of new examples of congruences of lines of order one, i.e. the congruences associated to the completely exceptional Monge-Ampère equations. We prove that they are in general not linear, and that through a general point of the focal locus there passes a planar pencil of lines of the congruence. In particular, the completely exceptional Monge-Ampère equations are of Temple type.

In the note we give an uniform bound for the multiple point Seshadri constants on $\mathbb{P}^1\times\mathbb{P}^1$.

We compute the class of the compactified Hurwitz divisor $\overline{\mathfrak{TR}}_d$ in $\overline{\mathcal{M}}_{2d-3}$ consisting of curves of genus $g=2d-3$ having a pencil $\mathfrak g^1_d$ with two unspecified triple ramification points. This is the first explicit example of a geometric divisor on $\overline{\mathcal{M}}_g$ which is not pulled-back form the moduli space of pseudo-stable curves. We show that the intersection of $\overline{\mathfrak{TR}}_d$ with the boundary divisor $\Delta_1$ in $\overline{\mathcal{M}}_g$ picks-up the locus of Fermat cubic tails.

It is an open problem to determine the Hilbert function and graded Betti numbers for the ideal of a fat point subscheme supported at general points of the projective plane. In fact, there is not yet even a general explicit conjecture for the graded Betti numbers. Here we formulate explicit asymptotic conjectures for both problems. We work over an algebraically closed field $K$ of arbitrary characteristic.

We show that there are no rational curves of degree $d$, $2\leq d\leq 15$, on a general heptic hypersurface in $\mathbb{P}^5$.

We describe an approach for computing arbitrarily accurate estimates for multi-point Seshadri constants for $n$ generic points of ${\bf P}^{2}$. We apply the approach to obtain improved estimates. We work over an algebraically closed field of characteristic 0.

We relate the minimal gonality of smooth curves in a complete, ample, base point free linear system $|L|$ on an Enriques surface to the existence of certain secant spaces on the image of the surface mapped by the adjoint system. We also explicitly compute the minimal gonality in terms of invariants of the line bundle $L$. In particular, we obtain a precise criterion for the variation of the gonality of the curves.

In this note we consider multiples $aD$, where $D$ is a divisor of the blow-up of $\mathbb{P}^n$ along points in general position which appears in the Alexander and Hirschowitz list of Veronese embeddings having defective secant varieties. In particular we show that there is such a $D$ with $h^1(X,D) > 0$ and $h^1(X,2D) = 0$.

A characterization of nef and good divisors is given: a divisor $D$ on a smooth complex projective variety is nef and good if and only if the asymptotic multiplier ideals of sufficiently high multiples of $e(D)\cdot D$ are trivial, where $e(D)$ denotes the exponent of the divisor $D$. Some results of the same kind are proved in the analytic setting.

In [11] we showed that if the multiple point Seshadri constants of an ample line bundle on a smooth projective surface in very general points satisfy certain inequality then the surface is fibred by curves computing these constants. Here we characterize the border case of polarized surfaces whose Seshadri constants in general points fulfill the equality instead of inequality and which are not fibred by Seshadri curves. It turns out that these surfaces are the projective plane and surfaces of minimal degree.