Related papers: On a question of Dolgachev

Let N_d be the number of degree d, nodal, rational plane curves through 3d-1 points in the complex projective plane. The number of degree d>=3, nodal, elliptic plane curves with a fixed (general) j-invariant through 3d-1 points is found to…

We prove orientation results for evaluation maps of moduli spaces of rational stable maps to del Pezzo surfaces over a field, both in characteristic $0$ and in positive characteristic. These results and the theory of degree developed in a…

In this note, we consider rational cuspidal plane curves having exactly one cusp whose complements have logarithmic Kodaira dimension two. We classify such curves with the property that the strict transforms of them via the minimal embedded…

We show the Kontsevich space of rational curves of degree at most roughly $\frac{2-\sqrt{2}}{2}n$ on a general hypersurface $X\subset \mathbb{P}^n$ of degree $n-1$ is equidimensional of expected dimension and has two components: one…

Let $M$ be the moduli space of rank 3 stable bundles with fixed determinant of degree 1 on a smooth projective curve of genus $g\geq 2$. When $C$ is generic, we show that any essential elliptic curve on $M$ has degree (respect to…

We make explicit computations in the formal symplectic geometry of Kontsevich and determine the Euler characteristics of the three cases, namely commutative, Lie and associative ones, up to certain weights.From these, we obtain some…

We study the geometry of spaces of planes on smooth complete intersections of three quadrics, with a view toward rationality questions.

We determine all modular curves $X_0(N)$ with infinitely many quartic points. To do this, we define a pairing that induces a quadratic form representing all possible degrees of a rational morphism from $X_0(N)$ to a positive rank elliptic…

On a general hypersurface of degree $d\leq n$ in $\mathbb P^n$ or $\mathbb P^n$ itself, we prove the existence of curves of any genus and high enough degree depending on the genus passing through the expected number $t$ of general points or…

We give a conjectural formula for the characteristic number of rational cuspidal curves in the projective plane by extending the idea of Kontsevich's recursion formula (namely, pulling back the equality of two divisors in the four pointed…

The aim of the present paper is to prove the rationality of the universal family of polarized $ K3 $ surfaces of degree 14. This is achieved by identifying it with the moduli space of cubic fourfolds plus the data of a quartic scroll. The…

The surface corresponding to the moduli space of quadratic endomorphisms of $\mathbb{P}^1$ with a marked periodic point of order $n$ is studied. It is shown that the surface is rational over $\mathbb{Q}$ when $n\le 5$ and is of general type…

When the plane is pie-sliced in $n\leq 4$ parts (with nonempty interior and common vertex at the origin) our main result provides a sufficient condition for any map $L$, that is continuous and piecewise linear relatively to this slicing, to…

We extend the formulae of classical invariant theory for the Jacobian of a genus one curve of degree $n \le 4$ to curves of arbitrary degree. To do this, we associate to each genus one normal curve of degree $n$, an $n \times n$ alternating…

Let $\L_g^G$ denote the locus of hyperelliptic curves of genus $g$ whose automorphism group contains a subgroup isomorphic to $G$. We study spaces $\L_g^G$ for $G \iso \Z_n, \Z_2{\o}\Z_n, \Z_2{\o}A_4$, or $SL_2(3)$. We show that for $G \iso…

The non-uniqueness of a rational parametrization of a rational plane curve may influence the process of computing envelopes of 1-parameter families of plane curves. We study envelopes of family of circles centred on a regular trifolium and…

We prove the irreducibility of moduli spaces of rational curves on a general del Pezzo threefold of Picard rank $1$ and degree $1$. As corollaries, we confirm Geometric Manin's conjecture and enumerativity of certain Gromov-Witten…

This note is motivated by the Question 16 of http://cubics.wikidot.com: Which configurations of 15 points in the projective 3-space arise as eigenpoints of a cubic surface? We prove that a general eigenscheme in the projective n-space is…

We show that $\mathcal{M}_{g,n}$, the moduli space of smooth curves of genus $g$ together with $n$ marked points, is unirational for $g=12$ and $2 \leq n\leq 4$ and for $g=13$ and $1 \leq n \leq 3$, by constructing suitable dominant…

We answer some enumerative questions about irreducible rational curves on Hirzebruch surfaces, by combining an idea of Kontsevich with the study of the geometry of certain natural parameter spaces. Our formulas generalize Kontsevich's…