Rutgers University

Department of Mathematics

Nicola Tarasca



Papers


  1. Classes of genus two curves with Weierstrass points (with Renzo Cavalieri)

    In preparation

    The locus of curves of genus two with n marked Weiestrass points has codimension n inside the moduli space of genus two, n-pointed curves, for 1<=n<=6. We produce an explicit formula for the classes of these loci. The formula is expressed using a generating function over stable graphs indexing the boundary strata of moduli spaces of pointed stable curves.

  2. K-classes of Brill-Noether loci and a determinantal formula (with Dave Anderson and Linda Chen)

    Submitted, arXiv:1705.02992

    We prove a determinantal formula for the K-theory class of certain degeneracy loci, and apply it to compute the Euler characteristic of the structure sheaf of the Brill-Noether locus of linear series with special vanishing at marked points. When the Brill-Noether number \rho is zero, we recover the Castelnuovo formula for the number of g^r_d's on a general curve; when \rho=1, we recover the formulas of Eisenbud-Harris, Pirola, and Chan-López-Pflueger-Teixidor for the arithmetic genus of a Brill-Noether curve of special divisors.

    Our degeneracy locus formula also specializes to new determinantal expressions for the double Grothendieck polynomials corresponding to 321-avoiding permutations, and gives double versions of the flagged skew Grothendieck polynomials recently introduced by Matsumura. Our result extends the formula of Billey-Jockusch-Stanley expressing Schubert polynomials for 321-avoiding permutations as generating functions for skew tableaux.

  3. Du Val curves and the pointed Brill-Noether Theorem (with Gavril Farkas)

    Selecta Mathematica, 23:3 (2017), pp. 2243-2259

    We show that a general curve in an explicit class of what we call Du Val pointed curves satisfies the Brill-Noether Theorem for pointed curves. Furthermore, we prove that a generic pencil of Du Val pointed curves is disjoint from all Brill-Noether divisors on the universal curve. This provides explicit examples of smooth pointed curves of arbitrary genus defined over Q which are Brill-Noether general. A similar result is proved for 2-pointed curves as well using explicit curves on elliptic ruled surfaces.

  4. Extremality of loci of hyperelliptic curves with marked Weierstrass points (with Dawei Chen)

    Algebra & Number Theory, 10:9 (2016), pp. 1935–1948

    The locus of genus-two curves with n marked Weierstrass points has codimension n inside the moduli space of genus-two curves with n marked points, for n<=6. It is well known that the class of the closure of the divisor obtained for n=1 spans an extremal ray of the cone of effective divisor classes. We generalize this result for all n: we show that the class of the closure of the locus of genus-two curves with n marked Weierstrass points spans an extremal ray of the cone of effective classes of codimension n, for n<=6. A related construction produces extremal nef curve classes in moduli spaces of pointed elliptic curves.

  5. Loci of Curves with Subcanonical Points in Low Genus (with Dawei Chen)

    Mathematische Zeitschrift, 284:3 (2016), pp. 683–714

    Inside the moduli space of curves of genus three with one marked point, we consider the locus of hyperelliptic curves with a marked Weierstrass point, and the locus of non-hyperelliptic curves with a marked hyperflex. These loci have codimension two. We compute the classes of their closures in the moduli space of stable curves of genus three with one marked point. Similarly, we compute the class of the closure of the locus of curves of genus four with an even theta characteristic vanishing with order three at a certain point. These loci naturally arise in the study of minimal strata of Abelian differentials.

  6. Divisors of secant planes to curves

    Journal of Algebra, 454 (2016), pp. 1-13

    Inside the symmetric product of a very general curve, we consider the codimension-one subvarieties of symmetric tuples of points imposing exceptional secant conditions on linear series on the curve of fixed degree and dimension. We compute the classes of such divisors, and thus obtain improved bounds for the slope of the cone of effective divisor classes on symmetric products of a very general curve. By letting the moduli of the curve vary, we study more generally the classes of the related divisors inside the moduli space of stable pointed curves.

  7. Pointed Castelnuovo numbers (with Gavril Farkas)

    Mathematical Research Letters, 23:2 (2016), pp. 389-404.

    The classical Castelnuovo numbers count linear series of minimal degree and fixed dimension on a general curve, in the case when this number is finite. For pencils, that is, linear series of dimension one, the Castelnuovo specialize to the better known Catalan numbers. Using the Fulton-Pragacz determinantal formula for flag bundles and combinatorial manipulations, we obtain a compact formula for the number of linear series on a general curve having prescribed ramification at an arbitrary point, in the case when the expected number of such linear series is finite. The formula is then used to solve some enumerative problems on moduli spaces of curves.

  8. Double Total Ramifications for Curves of Genus 2

    International Mathematics Research Notices, 19 (2015), pp. 9569-9593

    Inside the moduli space of curves of genus 2 with 2 marked points we consider the loci of curves admitting a map to P^1 of degree d totally ramified over the two marked points, for d>=2. Such loci have codimension two. We compute the class of their compactifications in the moduli space of stable curves. Several results will be deduced from this computation.

  9. Brill-Noether Loci in Codimension Two

    Compositio Mathematica, 149:09 (2013), pp. 1535-1568

    Let us consider the locus in the moduli space of curves of genus 2k defined by curves with a pencil of degree k. Since the Brill-Noether number is equal to -2, such a locus has codimension two. Using the method of test surfaces, we compute the class of its closure in the moduli space of stable curves.

    Extended abstract for Oberwolfach talk, in Oberwolfach Reports, 10:1 (2013), pp. 343–392
  10. Double Points of Plane Models in M_{6,1}

    Journal of Pure and Applied Algebra, 216:4 (2012), pp. 766-774

    The aim of this paper is to compute the class of the closure of the effective divisor in M_{6,1} given by pointed curves [C,p] with a sextic plane model mapping p to a double point. Such a divisor generates an extremal ray in the pseudoeffective cone of M_{6,1} as shown by Jensen. A general result on some families of linear series with adjusted Brill-Noether number 0 or -1 is introduced to complete the computation.

  11. Geometric Cycles in Moduli Spaces of Curves

    PhD Thesis, Humboldt University in Berlin, 2012