Let n≥4, 2≤r≤n−2 and e≥1. We show that the intersection of the locus of degree e morphisms from P1 to G(r,n) with the restricted universal sub-bundles having a given splitting type and the locus of ...degree e morphisms with the restricted universal quotient-bundle having a given splitting type is non-empty and generically transverse along at least one component of the intersection. As a consequence, we get that the locus of degree e morphisms from P1 to G(r,n) with the restricted tangent bundle having a given splitting type need not always be irreducible.
We study rational curves on general Fano hypersurfaces in projective space, mostly by degenerating the hypersurface along with its ambient projective space to reducible varieties. We prove results on ...existence of low-degree rational curves with balanced normal bundle, and reprove some results on irreducibility of spaces of rational curves of low degree.
Complex rational curves have been used to represent circular splines as well as many classical curves including epicycloids, cardioids, Joukowski profiles, and the lemniscate of Bernoulli. Complex ...rational curves are known to have low degree (typically half the degree of the corresponding rational planar curve), circular precision, invariance with respect to Möbius transformations, special implicit forms, an easy detection procedure, and a fast algorithm for computing their μ-bases. But only certain very special rational planar curves are also complex rational curves. To construct a wider collection of curves with similar appealing properties, we generalize complex rational curves to hyperbolic and parabolic rational curves by invoking the hyperbolic and parabolic numbers. We show that the special properties of complex rational curves extend to these hyperbolic and parabolic rational curves. We also provide examples to flesh out the theory.
Let k be a field with algebraic closure k¯ and D⊂Pk¯1 a reduced, effective divisor of degree n≥3, write kD for the field of moduli of D. A. Marinatto proved that when n is odd, or n=4, D descends to ...a divisor on PkD1.
We analyze completely the problem of when D descends to a divisor on a smooth, projective curve of genus 0 on kD, possibly with no rational points. In particular, we study the remaining cases n≥6 even, and we obtain conceptual proofs of Marinatto's results and of a theorem by B. Huggins about the field of moduli of hyperelliptic curves.
The moduli space M‾0,n of n pointed stable curves of genus 0 admits an action of the symmetric group Sn by permuting the marked points. We provide a closed formula for the character of the Sn-action ...on the cohomology of M‾0,n. This is achieved by studying wall crossings of the moduli spaces of quasimaps which provide us with a new inductive construction of M‾0,n, equivariant with respect to the symmetric group action. Moreover we prove that H2k(M‾0,n) for k≤3 and H2k(M‾0,n)⊕H2k−2(M‾0,n) for any k are permutation representations. Our method works for related moduli spaces as well and we provide a closed formula for the character of the Sn-representation on the cohomology of the Fulton-MacPherson compactification P1n of the configuration space of n points on P1 and more generally on the cohomology of the moduli space M‾0,n(Pm−1,1) of stable maps.
Automorphisms of projective structures Falla Luza, Maycol; Loray, Frank
Journal of geometry and physics,
July 2024, 2024-07-00, Volume:
201
Journal Article
Peer reviewed
Open access
We study the problem of classifying local projective structures in dimension two having non trivial Lie symmetries. In particular we obtain a classification of foliated projective structures having ...positive dimensional Lie algebra of projective vector fields.
We study parameter spaces of linear series on projective curves in the presence of unibranch singularities, i.e. cusps; and to do so, we stratify cusps according to value semigroup. We show that ...generalized Severi varieties of maps P1→Pn with images of fixed degree and arithmetic genus are often reducible whenever n≥3. We also prove that the Severi variety of degree-d maps with a hyperelliptic cusp of delta-invariant g≪d is of codimension at least (n−1)g inside the space of degree-d holomorphic maps P1→Pn; and that for small g, the bound is exact, and the corresponding space of maps is the disjoint union of unirational strata. Finally, we conjecture a generalization for unicuspidal rational curves associated to an arbitrary value semigroup.
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