Integral Projective Curve Research Articles

Picard bundle on the moduli space of torsionfree sheaves

Let Y be an integral nodal projective curve of arithmetic genus $$g\ge 2$$ with m nodes defined over an algebraically closed field. Let n and d be mutually coprime integers with $$n\ge 2$$ and $$d > n(2g-2)$$. Fix a line bundle L of degree d on Y. We prove that the Picard bundle $$\mathbf{E}_L$$ over the ‘fixed determinant moduli space’ $$U_L(n,d)$$ is stable with respect to the polarisation $$\theta _L$$ and its restriction to the moduli space $$U'_L(n,d)$$, of vector bundles of rank n and determinant L, is stable with respect to any polarisation. There is an embedding of the compactified Jacobian $$\bar{J}(Y)$$ in the moduli space $$U_Y(n,d)$$ of rank n and degree d. We show that the restriction of the Picard bundle of rank ng (over $$U_Y(n,n(2g-1))$$) to $$\bar{J}(Y)$$ is stable with respect to any theta divisor $$\theta _{{\bar{J}}(Y)}$$.

A speciality theorem for curves in $${\mathbb {P}}^5$$ P 5 contained in Noether–Lefschetz general fourfolds

Let \(C\subset {\mathbb {P}}^r\) be an integral projective curve. We define the speciality index e(C) of C as the maximal integer t such that \(h^0(C,\omega _C(-t))>0\), where \(\omega _C\) denotes the dualizing sheaf of C. In the present paper we consider \(C\subset {\mathbb {P}}^5\) an integral degree d curve and we denote by s the minimal degree for which there exists a hypersurface of degree s containing C. We assume that C is contained in two smooth hypersurfaces F and G, with \(deg(F)=n>k=deg (G)\). We assume additionally that F is Noether–Lefschetz general, i.e. that the 2-th Neron–Severi group of F is generated by the linear section class. Our main result is that in this case the speciality index is bounded as \(e(C)\le {\frac{d}{snk}}+s+n+k-6.\) Moreover equality holds if and only if C is a complete intersection of \(T:=F\cap G\) with hypersurfaces of degrees s and \({\frac{d}{snk}}\).

3D and 2D face recognition using integral projection curves based depth and intensity images

This paper presents an automatic face recognition system in the presence of illumination, expressions and pose variations based on depth and intensity information. At first, the registration of 3D faces is achieved using iterative closest point ICP. Nose tip point must be located using Maximum Intensity Method. This point usually has the largest depth value; however there is a problem with some unnecessary data such as: shoulders, hair, neck and parts of clothes; to cope with this issue, we propose the integral projection curves IPC-based facial area segmentation to extract the facial area. After that, the combined method principal component analysis PCA with enhanced Fisher model EFM is used to obtain the feature matrix vectors. Finally, the classification is performed using distance measurement and support vector machine SVM. The experiments are implemented on two face databases CASIA3D and GavabDB; our results show that the proposed method achieves a high recognition performance.

Generalized parabolic Hitchin pairs

We define generalized parabolic Hitchin (GPH) pairs on a non-singular curve X and construct their moduli spaces. When X is the normalization of an integral projective curve Y, we give a birational morphism from the moduli space of good GPH on X to the moduli space of Hitchin pairs on Y. We define a Hitchin map on GPH and show that it is a proper map on the moduli space of GPH and induces a proper Hitchin morphism on a subscheme of the moduli space of Hitchin pairs on Y. We study the relationship between Hitchin pairs and the compactified Jacobian of a spectral curve.

SINGULAR CURVES WITH LINE BUNDLES L DEFINED OVER ${\mathbb F}_q$ AND WITH H0(L) = H1(L) = 0

Here we prove the existence of several pairs (X, L), where X is a geometrically integral projective curve defined over 𝔽q and L is a line bundle on X defined over 𝔽q and with H0(X, L) = H1(X, L) = 0. These examples are obtained using the existence of similar line bundles on the normalization of X, i.e. a case studied by C. Ballet, C. Ritzenthaler and R. Roland.

Autoduality of compactified Jacobians for curves with plane singularities

Let C C be an integral projective curve with at most planar singularities. Consider its Jacobian J J and the compactified Jacobian J ¯ \overline {J} . We construct a flat family P ¯ \overline {P} of Cohen-Macaulay sheaves on J ¯ \overline {J} parametrized by J ¯ \overline {J} ; its restriction to J × J ¯ J\times \overline {J} is the Poincaré line bundle. We prove that the Fourier-Mukai transform given by P ¯ \overline {P} is an auto-equivalence of the derived category of J ¯ \overline {J} .

Refining Castelnuovo-Halphen bounds

Fix integers r,d,s,π with r≥4, d≫s, r−1≤s≤2r−4, and π≥0. Refining classical results for the genus of a projective curve, we exhibit a sharp upper bound for the arithmetic genus pa(C) of an integral projective curve C⊂ℙr of degree d, assuming that C is not contained in any surface of degree π. Next we discuss other types of bound for pa(C), involving conditions on the entire Hilbert polynomial of the integral surfaces on which C may lie.

Cohomology of line bundles on compactified Jacobians

Let C be an integral projective curve with surficial singularities. We prove that topologically trivial line bundles on the compactified Jacobian of C are in one-to-one correspondence with line bundles on C (the autoduality conjecture), and compute the cohomology of the line bundles. We also show that the natural Fourier-Mukai functor between the derived categories of quasi-coherent sheaves on the Jacobian and on the compactified Jacobian is fully faithful.

A speciality theorem for curves in P5

Let \(C{\subset}\,{\bf P}^r\) be an integral projective curve. One defines the speciality index e(C) of C as the maximal integer t such that \(h^0(C,\omega_C(-t)) > 0\) , where ωC denotes the dualizing sheaf of \(C\) . Extending a classical result of Halphen concerning the speciality of a space curve, in the present paper we prove that if \(C {\subset}\,{\bf P}^5\) is an integral degree d curve not contained in any surface of degree > }s{ > > } t > > u\geq 1\) , then \( e(C)\leq {\frac{d}{s}}+{\frac{s}{t}}+{\frac{t}{u}}+u-6. \) Moreover equality holds if and only if C is a complete intersection of hypersurfaces of degrees u, \({\frac{t}{u}}\) , \({\frac{s}{t}}\) and \({\frac{d}{s}}\) . We give also some partial results in the general case \(C\subset {\bf P}^r\) , \(r\geq 3\) .

The compactified Picard scheme of the compactified Jacobian

Let C be an integral projective curve in any characteristic. Given an invertible sheaf L on C of degree 1, form the corresponding Abel map A L : C → J ¯ , which maps C into its compactified Jacobian, and form its pullback map A L * : Pic J ¯ 0 → J , which carries the connected component of 0 in the Picard scheme back to the Jacobian. If C has, at worst, double points, then A L * is known to be an isomorphism. We prove that A L * always extends to a map between the natural compactifications, Pic J ¯ ÷ → J ¯ , and that the extended map is an isomorphism if C has, at worst, ordinary nodes and cusps.

A brill - noether theory forκ-gonal nodal curves

LetV(g, x, k, y) be the set of all pairs (X, F), whereX is an integral projective nodal curve withpa(X)=g and card(Sing(X))=x andF is a rank 1 torsion free sheaf onX with deg(F)=k, card(Sing(F))=y andh0(X, F)≥2. Here we study a general (X, F) eV(g, x, k, y) and in particular the Brill-Noether theory ofX and the scrollar invariants ofF.

Autoduality of the Compactified Jacobian

The following autoduality theorem is proved for an integral projective curve C in any characteristic. Given an invertible sheaf L of degree 1, form the corresponding Abel map AL:C→, which maps C into its compactified Jacobian, and form its pullback map , which carries the connected component of 0 in the Picard scheme back to the Jacobian. If C has, at worst, points of multiplicity 2, then is an isomorphism, and forming it commutes with specializing C.Much of the work in the paper is valid, more generally, for a family of curves with, at worst, points of embedding dimension 2. In this case, the determinant of cohomology is used to construct a right inverse to . Then a scheme‐theoretic version of the theorem of the cube is proved, generalizing Mumford's, and it is used to prove that is independent of the choice of L. Finally, the autoduality theorem is proved. The presentation scheme is used to achieve an induction on the difference between the arithmetic and geometric genera; here, special properties of points of multiplicity 2 are used.

Linear series onk-gonal curves

Here we prove the following result. Theorem 1.1.Let X be an integral projective curve of arithmetic genus g and k≧ ≧4 an integer. Assume the existence of L ∈ Pick (X) with h 0 (X, L)=2 and L spanned. Fix a rank 1 torsion free sheaf M on X with h 0(X,M)=r+1≧2, h1 (X, M)≧2 and M spanned by its global sections. Set d≔deg(M) and s≔max {n≧0:h 0 (X, M ⊗(L*)⊗n)>0}. Then one of the following cases occur: We find also other upper bounds onh 0 (X, F).

The multiplication map for global sections of line bundles and rank 1 torsion free sheaves on curves

LetX be an integral projective curve andL ∃ Pica(X),M ∃ Picb (X) with h1(X, L)= h1(X, M) = 0 andL, M general. Here we study the rank of the multiplication map μ L,M :H 0(X,L)⊗H 0(X,M)→H 0(X,L⊗M). We also study the same problem whenL andM are rank 1 torsion free sheaves onX. Most of our results are forX with only nodes as singularities.

BRILL–NOETHER THEORY OF CURVES WITH Ak SINGULARITIES AND HYPERPLANE SECTIONS OF K3 SURFACES

Let Y be an integral projective curve whose singularities are of type Ak, i.e. with only tacnodes and planar (perhaps non-ordinary) cusps. Set g:= pa(Y). Here we study the Brill - Noether theory of spanned line bundles on Y. If the singularities are bad enough, we show the existence of spanned degree d line bundles, L, with h0(Y, L) ≥ r + 1 even if the Brill - Noether number ρ(g, d, r) < 0. We apply this result to prove that genus g curves with certain singularities cannot be hyperplane section of a simple K3 surface S ⊂ P g.

Maximal degree subsheaves of torsion free sheaves on singular projective curves

Fix integers r, k, g with r > k > 0 and g ≥ 2. Let X be an integral projective curve with g := pa(X) and E a rank r torsion free sheaf on X which is a flat limit of a family of locally free sheaves on X. Here we prove the existence of a rank k subsheaf A of E such that r(deg(A)) ≥ k(deg(E)) − k(r − k)g. We show that for every g ≥ 9 there is an integral projective curve X,X not Gorenstein, and a rank 2 torsion free sheaf E on X with no rank 1 subsheaf A with 2(deg(A)) ≥ deg(E) − g. We show the existence of torsion free sheaves on non-Gorenstein projective curves with other pathological properties.

Erratum to the paper “generalised parabolic sheaves on an integral projective curve”

Erratum to the paper “generalised parabolic sheaves on an integral projective curve”

Generalized parabolic sheaves on an integral projective curve

We extend the notion of a parabolic vector bundle on a smooth curve to define generalized parabolic sheaves (GPS) on any integral projective curve X. We construct the moduli spacesM(X) of GPS of certain type onX. IfX is obtained by blowing up finitely many nodes inY then we show that there is a surjective birational morphism from M(X) to M (Y). In particular, we get partial desingularisations of the moduli of torsion-free sheaves on a nodal curveY.