Related papers: Clifford indices for vector bundles on curves

We study different notions of slope of a vector bundle over a smooth projective curve with respect to ampleness and affineness in order to apply this to tight closure problems. This method gives new degree estimates from above and from…

The Hilbert scheme of n points in the projective plane parameterizes degree n zero-dimensional subschemes of the projective plane. We examine the dual cones of effective divisors and moving curves on the Hilbert scheme. By studying…

We construct vector-valued modular forms on moduli spaces of curves and abelian varieties using effective divisors in projectivized Hodge bundles over moduli of curves. Cycle relations tell us the weight of these modular forms. In…

In this paper we obtain an explicit formula for the number of curves in a compact complex surface $X$ (passing through the right number of generic points), that has up to one node and one singularity of codimension $k$, provided the total…

We characterize the first min-max width of real projective spaces of any dimension. The width is the minimum area over the Clifford hypersurfaces. We also compute the Morse index of the Clifford hypersurfaces in the complex and quaternionic…

We study sections of the relative Picard bundle of a family of curves of genus $g \geq 2$ through the rank of the associated normal function. Using Griffiths' formula for the infinitesimal invariant and higher Schiffer variations, we…

Each vector space that is endowed with a quadratic form determines its Clifford algebra. This algebra, in turn, contains a distinguished group, known as the Lipschitz group. We show that only a quotient of this group remains meaningful in…

In the first part we use Gromov's K--area to define the K--area homology which stabilizes into singular homology on the category of pairs of compact smooth manifolds. The second part treats the questions of certain curvature gaps. For…

We prove new slope inequalities for relatively minimal fibred surfaces, showing an influence of the relative irregularity, of the unitary rank and of the Clifford index on the slope. The argument uses Xiao's method and a new Clifford-type…

Clifford algebras are important structures in Geometric Algebra and Quantum Mechanics. They have allowed a formalization of the primitive operators in Quantum Theory. The algebras are built over vector spaces with dimension a power of 2…

Let $C$ be a comb-like curve over $\mathbb{C}$, and $E$ be a vector bundle of rank $n$ on $C$. In this paper, we investigate the criteria for the semistability of the restriction of $E$ onto the components of $C$ when $E$ is given to be…

We classify invariant curves for birational surface maps that are expanding on cohomology. When the expansion is exponential, the arithmetic genus of an invariant curve is at most one. This implies severe constraints on both the type and…

We discuss natural connections between three objects: quadratic forms with values in line bundles, conic bundles and quaternion orders. We use the even Clifford algebra, and the Brauer-Severi Variety, and other constructions to give natural…

We compute Joyce's (arXiv:2111.04694) enumerative invariants $[\mathcal{M}^{\mathrm{ss}}_{(r,d)}]_{\mathrm{inv}}$ for semistable rank $r$ degree $d$ coherent sheaves on a complex projective curve. These invariants are a generalization of…

We systematically study the splitting of vector bundles on a smooth, projective variety, whose restriction to the zero locus of a regular section of an ample vector bundle splits. First, we find ampleness and genericity conditions which…

Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard…

We treat the problem of completing the moduli space for roots of line bundles on curves. Special attention is devoted to higher spin curves within the universal Picard scheme. Two new different constructions, both using line bundles on…

In this paper we obtain an explicit formula for the number of degree d curves in two dimensional complex projective space, passing through (d(d+3)/2 -k) generic points and having a codimension k singularity, where k is at most 7. In the…

We study sheaves E on a smooth projective curve X which are minimal with respect to the property that $h^0(E \otimes L) >0$ for all line bundles L of degree zero. We show that these sheaves define ample divisors D(E) on the Picard torus…

In this paper we obtain an explicit formula for the number of curves in two dimensional complex projective space, of degree d, passing through d(d+3)/2-(k+1) generic points and having one node and one codimension k singularity, where k is…