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\n Let\n \n \n \n \n U<\/mml:mi>\n \n \u2286\n \n <\/mml:mo>\n \n H<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msup>\n (<\/mml:mo>\n \n \n \n O<\/mml:mi>\n <\/mml:mrow>\n \n \n \n P<\/mml:mi>\n <\/mml:mrow>\n 1<\/mml:mn>\n <\/mml:msup>\n \n \u00d7\n \n <\/mml:mo>\n \n \n P<\/mml:mi>\n <\/mml:mrow>\n 1<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n (<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n )<\/mml:mo>\n <\/mml:mrow>\n U \\subseteq H^0({\\mathcal {O}_{\\mathbb {P}^1 \\times \\mathbb {P}^1}}(2,1))<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a basepoint free four-dimensional vector space. The sections corresponding to\n \n \n \n U<\/mml:mi>\n U<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n determine a regular map\n \n \n \n \n \n \n \u03d5\n \n <\/mml:mi>\n U<\/mml:mi>\n <\/mml:msub>\n :<\/mml:mo>\n \n \n \n P<\/mml:mi>\n <\/mml:mrow>\n 1<\/mml:mn>\n <\/mml:msup>\n \n \u00d7\n \n <\/mml:mo>\n \n \n P<\/mml:mi>\n <\/mml:mrow>\n 1<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n \n \u27f6\n \n <\/mml:mo>\n \n \n P<\/mml:mi>\n <\/mml:mrow>\n 3<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n \\phi _U: {\\mathbb {P}^1 \\times \\mathbb {P}^1} \\longrightarrow \\mathbb {P}^3<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We study the associated bigraded ideal\n \n \n \n \n \n I<\/mml:mi>\n U<\/mml:mi>\n <\/mml:msub>\n \n \u2286\n \n <\/mml:mo>\n k<\/mml:mi>\n [<\/mml:mo>\n s<\/mml:mi>\n ,<\/mml:mo>\n t<\/mml:mi>\n ;<\/mml:mo>\n u<\/mml:mi>\n ,<\/mml:mo>\n v<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n I_U \\subseteq k[s,t;u,v]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n from the standpoint of commutative algebra, proving that there are exactly six numerical types of possible bigraded minimal free resolution. These resolutions play a key role in determining the implicit equation for\n \n \n \n \n \n \n \u03d5\n \n <\/mml:mi>\n U<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n \n \n \n P<\/mml:mi>\n <\/mml:mrow>\n 1<\/mml:mn>\n <\/mml:msup>\n \n \u00d7\n \n <\/mml:mo>\n \n \n P<\/mml:mi>\n <\/mml:mrow>\n 1<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n \\phi _U({\\mathbb {P}^1 \\times \\mathbb {P}^1})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , via work of Bus\u00e9-Jouanolou, Bus\u00e9-Chardin, Botbol and Botbol-Dickenstein-Dohm on the approximation complex\n \n \n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {Z}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . In four of the six cases\n \n \n \n \n I<\/mml:mi>\n U<\/mml:mi>\n <\/mml:msub>\n I_U<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n has a linear first syzygy; remarkably from this we obtain all differentials in the minimal free resolution. In particular, this allows us to explicitly describe the implicit equation and singular locus of the image.\n <\/p>","DOI":"10.1090\/s0025-5718-2013-02764-0","type":"journal-article","created":{"date-parts":[[2013,8,14]],"date-time":"2013-08-14T11:48:48Z","timestamp":1376480928000},"page":"1337-1372","source":"Crossref","is-referenced-by-count":9,"title":["Syzygies and singularities of tensor product surfaces of bidegree (2,1)"],"prefix":"10.1090","volume":"83","author":[{"given":"Hal","family":"Schenck","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexandra","family":"Seceleanu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Javid","family":"Validashti","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2013,8,14]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1016\/S0022-4049(99)00100-0","article-title":"Bigeneric initial ideals, diagonal subalgebras and bigraded Hilbert functions","volume":"150","author":"Aramova, Annetta","year":"2000","journal-title":"J. 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