Search Results

I believe such bound exist. Perhaps you can deduce it from effective versions of Hilbert's Nullstellensatz. Take a look at Kollár's paper "Sharp Effective Nullstellensatz". The wikipedia page on Hilbe …

Not in general. Let $F \in H^0(\mathbb P^3, \mathcal O_{\mathbb P^3}(3))$ be a general cubic form and let $H \in H^0(\mathbb P^3, \mathcal O_{\mathbb P^3}(1))$ be a linear form. The kernel of $\omega …

Theorem 2.33 of Moduli of curves by Harris and Morrison gives a construction with weaker bounds. The idea is to start with a curve $C$ of genus at least two and consider the family of degree 3 branche …

I don't think so. Consider the product of two isogeneous elliptic curves. This surface has infinitely many smooth elliptic fibrations. The basis are all isogeneous but I guess that they belong to infi …

There exists singular Fano varieties of dimension $n\ge 3$ in characteristic $p$ ($p$ small when compared to $n$) with a non-zero section of $\Omega^{n-1}_X\otimes \mathcal L^*$ for an ample $\mathcal …

Yes for $G=\mathbb Z$, see Theorem 4.1 of this paper by Amerik-Campana.

A foliation on a $\mathbb P^1$-bundle over a curve of genus $g$ everywhere transverse to the fibers determines, and is completely determined, by its monodromy representation. The existence of a sectio …

If $X\subset \mathbb P^n$ is a smooth and non-degenerated variety then $$ \deg X \ge 1 + \mathrm{codim} X $$ as you can learn from Varieties of Minimal Degree by Eisenbud and Harris. Thus to under …

Let $X$ be a smooth complex variety of dimension $3$ and $Y$ a (perhaps singular) normal complex variety also of dimension $3$ which is smooth outside a point $y \in Y$. If needed one may assume th …

Let $C$ be a smooth projective curve contained in a smooth projective $3$-fold (everything defined over $\mathbb C$). Let $\widehat X$ be the formal completion of $X$ along $C$ and suppose there exist …

For a foliation on a projective manifold $Y$, the set of points belonging to invariant subschemes with a given Hilbert polynomial is a Zariski closed subset of $Y$. Therefore the set of algebraic leav …

Suppose $X$ is projective manifold endowed with holomorphic symplectic form. Let $D$ be smooth divisor on $X$. Characteristic foliation with algebraic and non-algebraic leaves. Suppose that $X$ is …

Let $\pi : X \to Y$ be a proper smooth morphism between complex quasi-projective varieties. Assume there exists a connected and Zariski dense analytic subvariety $Z$ of $Y$ such that for any two point …

There is also Simpson's proof that isolated points of the characteristic varieties of fundamental groups of projective manifolds are torsion. It also relies on Gelfond-Schneider Theorem. The moduli …

The foliation $\mathcal F$ defined by $p_1^* \omega_1 + p_2^* \omega_2$ is everywhere transverse to the fibration $p_2 : S \to C$. One can therefore lift paths from $C$ to leaves of $\mathcal F$ in …