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Irreducibility and Stable Rationality of the Loci of Curves of Genus at Most Six with a Marked Weierstrass Point

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Irreducibility and Stable Rationality of the Loci of Curves of Genus at Most Six with a Marked Weierstrass Point PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 142, Number 4, April 2014, Pages 1121–1132 S 0002-9939(2014)11899-5 Article electronically published on January 21, 2014 IRREDUCIBILITY AND STABLE RATIONALITY OF THE LOCI OF CURVES OF GENUS AT MOST SIX WITH A MARKED WEIERSTRASS POINT EVAN M. BULLOCK (Communicated by Lev Borisov) Abstract. Given a numerical semigroup H ⊆ (Z≥0, +), we consider the locus MH g,1 of smooth curves of genus g with a marked Weierstrass point of semi- ≤ MH group H. We show that for all semigroups H of genus g 6thelocus g,1 is irreducible and that for all but possibly two such semigroups it is stably rational. 1. Introduction By a curve, we will mean a projective curve over the complex numbers. Given a smooth connected curve C of genus g>1andapointp ∈ C,theWeierstrass semigroup of C at p is the set H = {n ∈ Z≥0 : some meromorphic function f has a pole of order n at p and no other poles}. It follows from the Riemann-Roch theorem that if H ⊆ Z≥0 is the Weierstrass semigroup of C at p,thenH consists of all but g of the positive integers, called the Weierstrass gaps.IfH ⊆ (Z≥0, +) is any subsemigroup with |Z≥0\H| = g,wesay that H is a numerical semigroup of genus g. For a general point p ∈ C, the Weierstrass semigroup is g +1,g+2,g+3,... and the gaps are 1, 2,...,g.AWeierstrass point on C is any point with a different semigroup. The pairs (C, p)wherep is a Weierstrass point of a smooth irreducible curve C form a divisor in the coarse moduli space Mg,1 of pointed curves. The MH ⊆M subset g,1 g,1 consisting of (C, p)wherep has semigroup H is locally closed. An irreducible variety V of dimension n is stably rational if V × Pm is birational to Pn+m for sufficiently large m; this condition is strictly stronger than unirational- ity, but strictly weaker than rationality. For example, stably rational varieties that are not rational are constructed in [BCTSSD85], and in [AM72] Artin and Mumford show that the torsion part of H3(X, Z) is a birational invariant of a smooth pro- jective variety X and construct unirational varieties of each dimension n ≥ 3 with non-trivial torsion in H3; such varieties cannot be stably rational by the K¨unneth theorem. Received by the editors June 29, 2011 and, in revised form, April 24, 2012 and May 8, 2012. 2010 Mathematics Subject Classification. Primary 14H45, 14H55; Secondary 14M20, 14E08, 14H10. c 2014 Evan M. Bullock 1121 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1122 EVAN M. BULLOCK In this paper, we prove the following: ≤ MH Theorem 1.1. Let H be a semigroup of genus g 6. Then the locus g,1 of smooth curves of genus g with a marked point of Weierstrass semigroup H is ir- MH reducible. Moreover, g,1 is always stably rational, with the possible exceptions of the semigroup 5, 7, 8, 9, 11 of a general Weierstrass point in genus 5 and the semigroup 6, 7, 8, 9, 10 of a general odd subcanonical point in genus 6. We will prove this in Section 2 after recalling the equivalence between the Weier- strass semigroup of C at p and the vanishing and ramification sequences of the canonical series KC at p and discussing previous results on the irreducibility of MH g,1. Weierstrass points can also be defined in terms of holomorphic 1-forms rather than meromorphic functions. Given a point p on a smooth curve C,thevanishing sequence (of the canonical series KC )atp is the ascending sequence 0 = a0(p) < a (p) < ···<a − (p) ≤ 2g − 2 of non-negative integers such that 1 g 1 0 {ai(p)} = vp(ω): ω ∈ H (C, KC ) ; i.e. the ai(p) are the zero orders at p of the global holomorphic 1-forms on C. It follows from the Riemann-Roch theorem (cf. [ACGH85], I.E) that the set of Weierstrass gaps at p is {ai(p)+1},soaWeierstrasspointisapointwherethe vanishing sequence is bigger than the sequence ai = i at a general point. The ramification sequence 0=α0 ≤ α1 ≤ ··· ≤ αg−1 ≤ g − 1 is the non- decreasing sequence defined by αi(p)=ai(p) − i; it also encodes the same infor- mation as the vanishing sequence or the semigroup. The weight of a Weierstrass point, g−1 w(p)= αi(p), i=0 measures how far a point is from being general. In particular, if H is a semigroup MH M of weight w, then every component of g,1 has codimension at most w in g,1;a component is called dimensionally proper if the codimension is equal to the weight. The most powerful tool for showing the existence of Weierstrass points for wide MH classes of numerical semigroups (i.e. the non-emptiness of g,1)isthemethod of limit linear series of Eisenbud and Harris [EH87a], which has since been used to prove several slight extensions of their results (e.g. [Kom91]), and was a major tool in showing that every numerical semigroup of genus at most 8 is a Weierstrass semigroup in [Kom94]. This method involves attaching an elliptic tail to a Weier- strass point on a genus g − 1 curve and smoothing to get a genus g curve. It is not typically well-suited for proving irreducibility because there is no easy way of MH M showing that the closure of a component of g,1 in g,1 would necessarily meet the boundary component Δ1,1. On the other hand, there are relatively few infinite classes of semigroups for which irreducibility is known (e.g. for ordinary Weierstrass points of weight 1 this was shown in [EH87b]). Weierstrass points with first non-gap 3 must lie on trigonal curves, and gap sequences of such points have been completely classified, including irreducibility for each gap sequence, in [BS98]. In [Arb74], Arbarello proves the irreducibility of the locus Wn,g of Weierstrass MH points whose first non-gap is n; this certainly shows that some g,1 is irreducible, but while we should expect H to be the “smallest” semigroup (with respect to License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use IRREDUCIBILITY OF WEIERSTRASS POINT LOCI IN LOW GENUS 1123 the natural partial order on the corresponding vanishing or ramification sequences) with this property, as far as I know this has never been proven. Similarly, in their study of the moduli space of pairs (C, ω)ofacurveanda holomorphic 1-form with a prescribed partition of zeros in [KZ03], Kontsevich and Zorich in particular analyze the case of subcanonical points, i.e. points p ∈ C ∼ such that KC = OC ((2g − 2) p), and show that this locus has three irreducible components for all g ≥ 4. In [Bul13], we show that each component is a different MH g,1 and determine for each genus the three semigroups, using a limit linear series argument as in [EH87a]. The general irreducibility theorems covering the most different non-trigonal semi- groups, however, are still those using earlier techniques. In [Pin74], Pinkham con- MH structed a compactification of g,1 for each numerical semigroup H = k1,...,kr by studying the weighted deformation theory of the singular monomial curve in Ar parameterized by (tk1 ,...,tkr ). In [Buc80, 4.2], Buchweitz noticed that applying some general unobstructedness results from deformation theory could be used to show the existence of Weierstrass points, in the cases where H has two or three generators, and the case where H has four generators and the ring C[tk1 ,...,tkr ]is Gorenstein. Of course if there are no obstructions, the Pinkham compactification MH of g,1 is just a weighted projective space, which is certainly irreducible, so the Buchweitz result implies irreducibility as well. The Gorenstein property in this case was shown to have a purely combinatorial description in [Kun70]: let c be the largest gap of H.WesaythatH is symmetric iff for all 0 ≤ i ≤ c,wehavethati is a gap if and only if c − i is a non-gap. Then C[tk1 ,...,tkr ] is Gorenstein if and only if H is symmetric. We thus have the following (the r ≤ 3 part of which was known to Nakano in [Nak08]): Theorem 1.2. Let H = k1,...,kr be a numerical semigroup. Suppose either that ≤ MH r 3 or that r =4and H is symmetric. Then g,1 is non-empty and irreducible and its Pinkham compactification is a weighted projective space. Remark. The Weierstrass points with symmetric semigroup are precisely the sub- canonical points discussed in [Bul13]. This follows from [Kun70, Lemma] since a point is subcanonical if and only if 2g − 1 is a gap. In [NM04] and [Nak08], Nakano and Mori used computer deformation theory MH calculations to prove irreducibility and rationality of g,1 for (almost) all the four- generator semigroups H of genus at most 6. In our proof of Theorem 1.1 in the next section, we use other methods to study the one remaining four-generator case and all the cases where H has more than four generators. 2. Proof of Theorem 1.1 By the results [NM04, Nak08] mentioned above, together with the results of Ballico, Casnati, and Fontanari [CF07,BCF09] showing that Mg,1 itself is rational ≤ MH for g 6, the irreducibility and rationality of g,1 is known for all semigroups of genus g ≤ 6 except the nine cases in Table 1.
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