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BrillNoether theory of binary curves
, 807
"... Abstract. The theorems of Riemann, Clifford and Martens are proved for every line bundle parametrized by the compactified Jacobian of every binary curve. The Clifford index is used to characterize hyperelliptic and trigonal binary curves. The BrillNoether theorem for r ≤ 2 is proved for a general b ..."
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Abstract. The theorems of Riemann, Clifford and Martens are proved for every line bundle parametrized by the compactified Jacobian of every binary curve. The Clifford index is used to characterize hyperelliptic and trigonal binary curves. The BrillNoether theorem for r ≤ 2 is proved for a general binary curve.
Naturality of Abel maps
"... Abstract. We give a combinatorial characterization of nodal curves admitting a natural (i.e. compatible with and independent of specialization) dth Abel map for any d ≥ 1. ..."
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Abstract. We give a combinatorial characterization of nodal curves admitting a natural (i.e. compatible with and independent of specialization) dth Abel map for any d ≥ 1.
The universal Whitham hierarchy and the geometry of the moduli space of pointed Riemann surfaces
, 2009
"... We show that certain structures and constructions of the Whitham theory, an essential part of the perturbation theory of soliton equations, can be instrumental in understanding the geometry of the moduli spaces of Riemann surfaces with marked points. We use the ideas of the Whitham theory to define ..."
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We show that certain structures and constructions of the Whitham theory, an essential part of the perturbation theory of soliton equations, can be instrumental in understanding the geometry of the moduli spaces of Riemann surfaces with marked points. We use the ideas of the Whitham theory to define local coordinates and construct foliations on the moduli spaces. We use these constructions to give a new proof of the Diaz’ bound on the dimension of complete subvarieties of the moduli spaces. Geometrically, we study the properties of meromorphic differentials with real periods, and their degenerations.
unknown title
, 2000
"... Abstract. In [3] D. Eisenbud and J. Harris posed the following question: What are the limits of Weierstrass points in families of curves degenerating to stable curves not of compact type? We answer their question for onedimensional families of smooth curves degenerating to stable curves with just t ..."
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Abstract. In [3] D. Eisenbud and J. Harris posed the following question: What are the limits of Weierstrass points in families of curves degenerating to stable curves not of compact type? We answer their question for onedimensional families of smooth curves degenerating to stable curves with just two components meeting at points in general position. In this note we treat only those families whose total space is regular. Nevertheless, we announce here our most general answer, to be presented in detail in [5]. 1. Regularly smoothable linear systems Let C be a connected, projective, nodal curve defined over an algebraically closed field k. Let C1,..., Cn be its irreducible components. Let B: = Spec(k[[t]]); let o denote its special point and η its generic point. A projective and flat map π: S → B is said to be a smoothing of C if the generic fiber Sη is smooth and the special fiber So is isomorphic to C. In addition, if S is regular then π is called a regular smoothing. If π: S → B is a regular smoothing, then C1,..., Cn are Cartier divisors on S, and C1 + · · · + Cn ≡ 0.
JETS OF SINGULAR FOLIATIONS
, 2006
"... Abstract. Given a singular foliation satisfying locally everywhere the Frobenius condition, even at the singularities, we show how to construct its global sheaves of jets. Our construction is purely formal, and thus applicable in a variety of contexts. 1. ..."
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Abstract. Given a singular foliation satisfying locally everywhere the Frobenius condition, even at the singularities, we show how to construct its global sheaves of jets. Our construction is purely formal, and thus applicable in a variety of contexts. 1.
ENRICHED SPIN CURVES ON STABLE CURVES WITH TWO COMPONENTS
, 810
"... Abstract. In [M], Mainò constructed a moduli space for enriched stable curves, by blowingup the moduli space of DeligneMumford stable curves. We introduce enriched spin curves, showing that a parameter space for these objects is obtained by blowingup the moduli space of spin curves. 1. ..."
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Abstract. In [M], Mainò constructed a moduli space for enriched stable curves, by blowingup the moduli space of DeligneMumford stable curves. We introduce enriched spin curves, showing that a parameter space for these objects is obtained by blowingup the moduli space of spin curves. 1.
Linear series on semistable curves.
, 2008
"... Abstract. For a semistable curve X of genus g, the number h 0 (X, L) is studied for line bundles L of degree d parametrized by the compactified Picard scheme. The theorem of Riemann is shown to hold. The theorem of Clifford is shown to hold in the following cases: X has two components; X is any semi ..."
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Abstract. For a semistable curve X of genus g, the number h 0 (X, L) is studied for line bundles L of degree d parametrized by the compactified Picard scheme. The theorem of Riemann is shown to hold. The theorem of Clifford is shown to hold in the following cases: X has two components; X is any semistable curve and d = 0 or d = 2g − 2; X is stable, free from separating nodes, and d ≤ 4. These results are shown to be sharp. Applications to the Clifford index, to the combinatorial description of hyperelliptic curves, and to plane quintics are given. Contents
RESEARCH STATEMENT
"... Although my research spans a wide range of topics in algebraic geometry, my various projects are nonetheless heavily interconnected. Broad themes include moduli spaces and deformation theory, geometric problems in positive characteristic, and new results on and applications of the EisenbudHarris th ..."
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Although my research spans a wide range of topics in algebraic geometry, my various projects are nonetheless heavily interconnected. Broad themes include moduli spaces and deformation theory, geometric problems in positive characteristic, and new results on and applications of the EisenbudHarris theory of limit linear series. Various aspects of my work touch on topics as diverse as modular towers and Shimura varieties in number theory, aspects of real and complex Schubert calculus, including degenerations of the Grassmannian, Hurwitz numbers and Ehrhart polynomials in combinatorics, and representations of quivers and of Sn. In a number of cases, results are obtained via shifts in perspective, or new connections between apparently unrelated fields. Examining each topic in turn, I describe highlights of my prior work, as well as my plans for the future. While there are almost certainly too many potential projects listed here for me to pursue all of them myself, I feel that several of them would constitute good thesis topics for graduate students. 1. Moduli spaces, stacks, and deformation theory Moduli spaces of curves, maps of curves, and vector bundles have come up in virtually of