Related papers: Projective varieties of maximal sectional regulari…

We consider a Cartier divisor L on a d-dimensional complex projective variety X. It is well-known that the dimensions of the cohomomology groups H^i(X,O_X(mL)) grow at most like m^d, and it is natural to ask when one of these actually has…

By a $B$-regular variety, we mean a smooth projective variety over $C$ admitting an algebraic action of the upper triangular Borel subgroup $B \subset SL_2(C)$ such that the unipotent radical in $B$ has a unique fixed point. A result of M.…

For a smooth projective variety $X$, we consider when the diagonal $\Delta_X$ is nef as a cycle on $X\times X$. In particular, we give a classification of complete intersections and smooth del Pezzo varieties where the diagonal is nef. We…

In this paper, we explain the regularity, projective dimension and depth of edge ideal of some classes of graphs in terms of invariants of graphs. We show that for a $C_5$-free vertex decomposable graph $G$, $\T{reg}(R/I(G))= c_G$, where…

We study a geometric version of the dimension growth conjecture. While it is closely related in spirit to themes arising in geometric Manin's conjecture, it applies in greater generality and provides more uniform bounds. For an irreducible…

For a projective variety $Z$ and for any integer $p$, define the $p$-th N\'eron-Severi group $NS_p(Z)$ of $Z$ as the image of the cycle map $A_{p}(Z)\to H_{2p}(Z; \mathbb{C})$. Now let $X\subset \Ps^{2m+1}$ ($m\geq 1$) be a projective…

Let $X$ be a smooth projective horospherical variety of Picard number one. We show that a uniruled projective manifold of Picard number one is biholomorphic to $X$ if its variety of minimal rational tangents at a general point is…

For an algebraic set $X$ (union of varieties) embedded in projective space, we say that $X$ satisfies property $\textbf{N}_{d,p}$, $(d\ge 2)$ if the $i$-th syzygies of the homogeneous coordinate ring are generated by elements of degree $<…

Fix an integer $n \geq 1$, and consider the set of all connected finite simple graphs on $n$ vertices. For each $G$ in this set, let $I(G)$ denote the edge ideal of $G$ in the polynomial ring $R = K[x_1,\ldots,x_n]$. We initiate a study of…

A curve of genus g is maximal Mumford (MM) if it has g+1 ovals and g tropical cycles. We construct full-dimensional families of MM curves in the Hilbert scheme of canonical curves. This rests on first-order deformations of graph curves…

Take a smooth, connected and non-degenerate projective curve $X\subset \mathbb {P}^r$, $r\ge 2b+2\ge 6$, defined over an algebraically closed field with characteristic $0$ and let $\sigma _b(X)$ be the $b$-secant variety of $X$. We prove…

We prove an effective bound for the degree of a smooth divisor of a hypersurface of P^n, n>4 (projective space over an algebraically closed field of characteristic zero). Our result follows from a strong (since the degree of the divisor is…

Maximum Variance Unfolding is one of the main methods for (nonlinear) dimensionality reduction. We study its large sample limit, providing specific rates of convergence under standard assumptions. We find that it is consistent when the…

Let X be a projective variety with terminal singularities and let L be an ample Cartier divisor on X. We prove that if f is a birational contraction associated to an extremal ray $ R \subset \bar {NE(X)}$ such that R.(K_X+(n-2)L)<0, then f…

Let $n=2,3,4,5$ and let $X$ be a smooth complex projective hypersurface of $\mathbb P^{n+1}$. In this paper we find an effective lower bound for the degree of $X$, such that every holomorphic entire curve in $X$ must satisfy an algebraic…

Multigraded Castelnuovo--Mumford regularity of a module $M$ over the total coordinate ring $S$ of a smooth projective toric variety $X$ is a region $\operatorname{reg} M \subset \operatorname{Pic} X$ invariant under translation by the nef…

The $X$-rank of a point $p$ in projective space is the minimal number of points of an algebraic variety $X$ whose linear span contains $p$. This notion is naturally submultiplicative under tensor product. We study geometric conditions that…

We investigate slicings of combinatorial manifolds as properly embedded co-dimension 1 submanifolds. A focus is given to dimension 3 where slicings are normal surfaces. In the case of 2-neighborly 3-manifolds and quadrangulated slicings, a…

Castelnuovo-Mumford regularity is a measure of algebraic complexity of an ideal. Regularity of monomial ideals can be investigated combinatorially. We use a simple graph decomposition and results from structural graph theory to prove,…

We study surjective (not necessarily regular) rational endomorphisms $f$ of smooth del Pezzo surfaces $X$. We prove that under certain natural non\,-\,degeneracy condition $f$ can have degree bigger than $1$ only when $(-K_X^2) > 5$. Some…